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Digitized  by  the  Internet  Archive 

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COURSE  IN 
DESCRIPTIVE  GEOMETRY  AND  STEREOTOMY. 

By  S.  EDWARD    WARREN,  C.  E. 

PROFESSOR   OF   DESCRIPTIVE    GEOMETRY,    ETC.  ;    FORMERLY    IN    THE    RENSSELAER    POLYTECHNIC 

INSTITUTE,    TROY. 


The  following  works,  published  successively  since  i860,  have  been  well  received  by  all  the  scien- 
tific and  educational  periodicals,  and  are  in  use  in  most  of  the  Engineering  and  Scientific  Schools  of 
the  country,  and  the  elementary  ones  in  many  of  the  Higher  Preparatory  Schools. 

The  Author,  by  his  long-continued  engagements  in  teaching,  has  enjoyed  facilities  for  the  prep- 
aration of  his  works  which  entitle  them  to  a  favorable  consideration. 


I.— ELEMENTARY    WORKS. 

These  are  designed  and  composed  with  great  care ;  primarily  for  the  use  of  all  higher  Public  and 
Private  Schools,  in  training  students  for  subsequent  professional  study  in  the  Engineering  and  Sci- 
entific Schools ;  then,  provisionally,  for  the  use  of  the  latter  institutions,  until  preparatory  training 
shall,  as  is  very  desirable,  more  generally  include  their  use ;  and,  finally,  for  the  self-instruction  of 
Teachers,  Artisans,  Builders,  etc. 

1.  ELEMENTARY  FREEHAND 

GEO  ME  TRICA  L  DRA  WING.  A  Series 
of  Progressive  Exercises  on  Regular  Lines  and 
Forms ;  including  Systematic  Instruction  in 
Lettering.  A  training  of  the  eye  and  hand  for 
all  who  are  learning  to  draw.  i2mo,  cloth, 
many  cuts,  75  cents. 

2.  ELEMENTARY  PLANE  PROB- 
LEMS. On  the  Point,  Straight  Line,  and 
Circle.  Division  I.  —  Preliminary  or  Instru- 
mental Problems.  Division  II.  —  Geometrical 
Problems.     i2mo,  cloth,  $1.25. 

3.  DRAFTING  INSTRUMENTS 
AND  OPERATIONS.  Division  I.  — In- 
struments and  Materials.  Division  II.  — Use 
of  Drafting  Instruments,  and  Representation 
of  Stone,  Wood,  Iron,  etc.  Division  III. — 
Practical  Exercises  on  Objects  of  Two  Dimen- 
sions (Pavements,  Masonry  Fronts,  etc.)  Di- 
vision IV.  —  Elementary  ^Esthetics  of  Geo- 
metrical Drawing.  One  volume,  i2mo,  cloth, 
51.25. 

Volumes  1  and  3  bound  together,  $1.75. 


4.  ELEMENTARY    PROJECTION 

DRA  WING.  Third  edition,  revised  and  en- 
larged. In  five  divisions.  I.  —  Projections  of 
Solids  and  Intersections.  II.  — Wood,  Stone, 
.  and  Metal  Details.  III.  —  Elementary  Shad- 
ows and  Shading.  IV.  —  Isometrical  and  Cab- 
inet Projections  (Mechanical  Perspective). 
V.  —  Elementary  Structures.  This  and  the 
last  volume  are  especially  valuable  to  all  Me- 
chanical Artisans.     i2mo,  cloth,  $1.50. 

5.  ELEMENTARY  LINEAR  PER- 
SPECTIVE OF  FORMS  AND  SHAD- 
OWS. With  many  Practical  Examples.  This 
volume  is  complete  in  itself,  and  differs  from 
many  other  elementary  works  in  clearly  dem- 
onstrating the  principles  on  which  tlie  prac- 
tical rules  0/  perspective  are  based,  without 
including  such  complex  problems  as  are  usu- 
ally found  in  higher  works  on  perspective.  It 
is  designed  especially  for  Young  Ladies'  Semi- 
naries, Artists,  Decorator',  and  Schools  of  De- 
sign, as  well  as  for  the  institutions  above  men- 
tioned.    One  volume,  i2mo,  cloth,  $1.00. 


II. -HIGHER  WORKS. 

These  are  designed  principally  for  Schools  of  Engineering  and  Architecture,  and  for  the  members 
generally  of  those  professions ;  and  the  first  three  also  for  all  Colleges  which  give  a  General  Scien- 
tific Course,  preparatory  to  the  fully  Professional  Study  of  Engineering,  etc. 


I.  DESCRIPTIVE        GEOMETRY. 

Adapted  to  Colleges  and  Liberal  Education, 
as  well  as  to  Technical  Schools  and  Technical 
Education .  Part  I.  —  Surfaces  of  Revolution. 
The  Point,  Line,  and  Plane,  Developable  Sur- 
faces, Cylinders  and  Cones,  and  the  Conic  Sec- 
tions, Warped  Surfaces,  the  Hyperboloid, 
Double-Curved  Surfaces,  the  Sphere,  Ellip- 
soid, Torus,  etc  ,  etc.  Complete  in  itself  by 
giving,  as  may  be  preferred,  Warped  Surfaces 
in  immediate  connection  with  their  applica- 
tions. One  volume,  8vo,  twenty-four  folding 
plates  and  wood-cuts,  cloth,  $4.00. 

II.  GENERAL      PROBLEMS      OF 

SHADES  AND  SHADOWS.  A  wide 
range  of  problems ;  gives  variety  without  rep- 
etition, and  a  thorough  discussion  of  the  prin- 


ciples of  Shading.  One  volume,  8vo,  with 
numerous  plates,  cloth,  $3.50. 

III.  HIGHER  LINEAR  PER- 
SPECTIVE. Containing  a  concise  sum- 
mary of  various  methods  of  perspective  con- 
struction ;  a  full  set  of  standard  problems  ;  and 
a  careful  discussion  of  special  higher  ones. 
With  numerous  plates.     8vo,  cloth,  $4.00. 

IT.    ELEMENTS      OF    MACHINE 

CONSTRUCTION  AND  DRAWING. 
On  a  new  plan,  and  enriched  by  many  stand- 
ard and  novel  examples  of  the  best  present 
practice.     Text  8vo.     Plates  4to.    $7.50. 

V.    STONE   CUTTING.    Cloth,  8vo,  ten 

plates,  §2.50. 


STERNOTOMY. 


PROBLEMS 


STONE    CUTTING. 


IN  FOUR  CLASSES. 


I.  —  PLANE-SIDED   STRUCTURES. 
II.  —STRUCTURES  CONTAINING  DEVELOPABLE  SURFACES. 

III.  —  STRUCTURES   CONTAINING  WARPED   SURFACES. 

IV.  —  STRUCTURES     CONTAINING    DOUBLE-CURVED    SUR- 

FACES. 


FOR  STUDENTS   OF  ENGINEERING  AND  ARCHITECTURE. 


S.  EDWARD  WARREN   C.  E. 

PROFESSOR  IN  THE   MASSACHUSETTS   NORMAL   ART   SCHOOL,  ETC.,  AND    FORMERLY    IN 
THE   RENSSELAER  POLYTECHNIC  INSTITUTE. 


NEW   YORK: 

JOHN   WILEY   AND   SON, 

15  Astor  Place. 

1875. 


Entered,  according  to  Act  of  Congress,  in  the  year  1875,  by 

S.  Edward  Warken,  C.  E., 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


RIVERSIDE,    CAMBRIDGE. 

8TEREOTYPED    AND    PRINTED     BY 

H.   O.    HOUGHTON   AND   COMPANY. 


TO  ALL  THOSE  GRADUATES  OF  THE 

KENSSELAER   POLYTECHNIC   INSTITUTE, 

IN    MANY    SUCCESSIVE    TEARS  J    WHO    DOUBTLESS    STILL    RETAIN    A    CLEAR    REC- 
OLLECTION, AND,    I    HOPE,   A    PLEASANT    REMEMBRANCE 
OP     THE     "OBLIQUE     ARCH,"     AND     THE 
"  COMPOUND   WING-WALL," 

AS    DRAWN    BY    THEM    UNDER    MY    INSTRUCTION  ; 

Cijts  little  Walnme, 

WHICH    CONTAINS    IMPROVED    ILLUSTRATIONS    OF    BOTH,    WITH    MANY    OTHER 
PROBLEMS,  IS    MOST    KINDLY   AND    VERY   RESPECTFULLY    DED- 
ICATED   BY    THEIR    FORMER    TEACHER    AND 
LASTING    FRIEND 

THE  AUTHOR. 


PKEFAOE. 


This  manual  has  been  composed  with  the  idea  of  represent- 
ing, essentially,  every  class  of  structures,  and  every  principal 
variety  of  surface,  so  as  to  make  it  most  widely  useful  to  the 
student  in  solving  any  other  problems  which  he  might  meet. 

The  student  needs,  desires,  and  appreciates  explicit  detailed 
information  in  due  abundance,  not  to  prevent  him  from  think- 
ing for  himself,  but  to  train  him  to  do  so  by  examples  fully 
explained.  On  this  principle,  and  supported  by  the  best  author- 
ities, I  have  discussed  the  few  problems  which  could  be  admit- 
ted within  the  proposed  limits,  so  thoroughly  as  to  satisfy,  I 
trust,  all  who  enjoy  the  most  —  indeed,  the  only  universally 
—  available  help,  viz.,  a  printed  text. 

I  have  attached  scales  and  dimensions  to  the  problems,  which 
teachers  and  students  may  use  or  not,  according  as  they  prefer 
to  work  as  if  drawing  actual  structures  for  practical  purposes, 
or  to  study  the  purely  geometrical  principles  and  operations 
involved.  In  either  case,  the  figures  should  be  made,  generally, 
from  two  to  three  times  as  large  as  those  of  the  plates  in  this 
volume,  carefully  following  the  text,  and  under  frequent  inter- 
rogation by  the  teacher,  in  doing  so.  I  should  add  that  this 
work  presupposes  a  fair  acquaintance  with  descriptive  geometry, 
though  many  of  its  problems  could  be  understood  after  the 
study  of  my  "  Elementary  Projection  Drawing."  It  is,  how- 
ever, complete  in  itself  in  regard  to  several  collateral  topics 
required  for  use  in  it,  and  not  as  conveniently  found  else- 
where. 

I  must  here  acknowledge  my  indebtedness  to  Leroy  for 
suggestions  of   practical  problems,  and  to  Adhemak,  in  per- 


VI  PREFACE. 

fecting  the  treatment  of  the  oblique  arch.  It  seemed  better  to 
take  examples  essentially  like  actual  structures,  rather  than 
imaginary  ones,  merely  for  the  sake  of  greater  apparent  origi- 
nality. But  I  have  in  every  case  made  such  changes,  in  vari- 
ous details  of  design  and  treatment,  including,  especially,  the 
novel  feature  of  numerous  examples  for  practice,  as  to  make 
my  volume  as  much  as  possible  a  new  contribution  as  well  as  a 
text-book.  Moreover,  problems  VIII.  and  XV.,  and  the  sys- 
tematic arrangement  presented  in  the  general  table,  will  not 
be  found  elsewhere. 

Newton,  Mass.,  May,  1875. 


NOTE   TO   TEACHERS. 


Two  complete  successive  courses,  elementary  and  higher,  can  be 
made  up  from  this  work,  as  follows  :  — 

Each  problem  to  be  drawn  in  Plan,  Elevation,  Section,  Details,  in 
isometric  or  oblique  projection,  and  Developments. 
I.  Elementary  Course.  —  a.  Plane-sided  structures.  —  1.  The  but- 
tressed walls  (Prob.  II.).     2.  The    plate  band  (Prob.    III.). 
3.  A  plane-sided  wing-wall  (Arts.  14-16). 

b.  Involving  developable  surfaces.  —  [See  Arts.  17-28,  and  Probs. 

IV.,  V.,  and  XVII.  (the  bracket),  with  their  examples.]  ■ — 
1.  The  segmental  —  2.  The  full  centred  —  3.  The  sloping 
front —  4.  The  skew  front —  5.  The  cylindrical-faced  — 
6.  The  rampant —  7.  The  conical  recessed,  arch.  8.  The 
trumpet  bracket. 

c.  Involving  warped  surfaces.  —  1.   Circular  stairs  around  a  central 

post.  [See  Arts.  119-125,  and  126.]  2.  The  warped-faced 
wall  (59). 

d.  Involving  double-curved  surfaces.  —  1.  The  Niche  (Prob.  XVII.) 

2.  The  dome  only  (Prob.  XIX.). 
II.  Higher  Course.  —  Any  selection  from  the  above,  as  introductory, 
for  those  who  have  not  previously  taken  the  foregoing  elementary 
course,  together  with  any  of  the  other  problems  of  each  of  the  four 
classes  noted  in  the  table  of  contents. 


V11L 


STONE-CUTTING. 


GEOMETRICAL   CLASSIFICATION. 
The  Characteristic  Surface. 


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CONTENTS. 


Preface .      v 

Note  to  Teachers vii 

STEREOTOMY :  Stone-cutting  :  First  Principles  ....       1 

CLASS  I. 

Plane-Sided  Structures. 

PROBLEM  I.  —  To  form  plane  surfaces  of  stone,  making  any  angle  with 

each  other 4 

PROBLEM  II.  —  A  sloping  wall,  and  truncated  pyramidal  buttress          .  5 

PROBLEM  III.  —  The  recessed  flat  arch,  or  plate-band         ....  8 

Plane-sided  Wing-Walls 10 

CLASS   II. 

Structures  containing  Developable  Surfaces. 

Arches.  —  Definitions 12 

Classification 13 

Preliminary  constructions 14 

§    Conic  Sections           ......  14 

1°.  —  To  construct  a  circle  by  points,  having  given  its  radius    .         .  14 
_2°. — To  construct  an  arc  of  a  circle  by  points,  knowing  its  chord 

and  versed-sine,  or  rise. 15 

3°.  —  To  construct  an  ellipse  by  points  on  given  axes.     Also,  nor- 
mals to  it  (two  methods) 15 

4°.  —  To  construct  the  arc  of  a  parabola  ;  on  a  given  segment  of  the 
axis,  and  a  chord  which  is  perpendicular  to  the  axis.    Also, 

normals  to  it                .         .         .         .         .         .         .         .  16 

5°.  —  To  construct  an  arc  of  a  hyperbola,  on  a  given  chord,  and  seg- 
ment of  the  axis,  perpendicular  to  the  chord    .         .         .  17 
§§  Poli/central  Arch  Curves       .         .         .         .         .18 

Three-Centred  Ovals      .         .         .         .         .         .         .         .         .  18 

1°.  —  To  construct  the  general  case  of  the  semi-oval  of  three  centres  18 
2°.  —  First  special  case.     The  semi-oval  of  three  centres  when  the 

lesser  arc  is  60° 18 

3°.  —  Second  special  case.     The  ratio  —  to  be  a  minimum  .         .         19 


X  CONTENTS. 

Five-Centred  Ovals 20 

4°.  —  To  construct  a  five-centred  semi-oval,  which  shall  conform  as 

nearly  as  possible  to  a  semi-ellipse,  on  the  same  axes  .     20 

5°.  —  To  construct  the  five-centred  oval,  by  a  method  applicable  to 

an  oval  having  any  number  of  centres  .        .         .         .21 

Illustrations. 

PEOBLEM  IV.  —  A  three-centred  arch  in  a  circular  wall      .         .         .         .22 

PEOBLEM  V.  —  A  semi-cylindrical  arch,  connecting  a  larger  similar  gal- 
lery, perpendicular  to  it,  on  the  same  springing  plane ;  with  an  en- 
closure which  terminates  the  arch  by  a  sloping  skew  face      .        .         .25 
Groined,  and  Cloistered  Arches      .......         29 

Theorem  I. — Having  two  cylinders  of  revolution,  whose  axes  intersect , 
the  projection  of  their  intersection,  upon  the  plane  of  their  axes,  is  a 

hyperbola 30 

PEOBLEM  VI.  — The  oblique  groined  arch' 32 

PEOBLEM  VII.  —  The  groined  and  cloistered,  or  elbow  arch      .        .         .35 
Conical,  or  Trumpet  Arches    ........         37 

PEOBLEM  VIII. — A  trumpet  in  the  angle  between  two  retaining  walls     .     37 
PEOBLEM  IX. — A  trumpet  arched  door,  on  a  corner      ....        40 

PEOBLEM  X.  —  An  arched  oblique  descent 44 

CLASS  III. 
Structures  containing  Warped  Surfaces. 
PEOBLEM  XL  —  The  recessed  Marseilles  gate  .         .         .         .         .         49 

The  Oblique  Arch. 

Preliminary  topics.  —  Elementary  mechanics  of  the  arch   .         .        .         .53 

The  resulting  standard,  or  essentially  perfect  design  for  an 

oblique  arch       .........     55 

PEOBLEM  XII.  —  The  partial,  and  trial  construction  of  the  orthogonal,  or 

equilibrated  arch 56 

The  Helix     .         .         .         .         .         .         .         .         .         .         .61 

Theorem  II.  —  The  projection  of  the  helix  on  a  plane  parallel  to  its  axis 

is  a  sinusoid .62 

The  Helicoid  ..........     62 

PEOBLEM  XIII. — A  segmental  oblique  arch,  on  the  helicoidal  system  .         63 

I.  The  Projections.     (Arts.  81-104)  .         . 63 

II.  The  Directing  Instruments        .         .         .         .         .         .         .         .         73 

III.  The  Application 76 

Useful  Numerical  Data     .........         79 

Modifications  of  the  Orthogonal  and  Helicoidal  Systems         .         .         .81 

Wing- Walls        .         .         ...         .         .83 

PEOBLEM  XIV.  —  The  compound,  or  piano-conical  wing-wall         .         .         85 

The  Conoid 92 

PEOBLEM  XV— The  conoidal  wing-wall 94 


CONTENTS.  XI 


Stairs  ......         97 


PKOBLEM  XVI.  —  Winding  stairs  on  an  irregular  ground  plan  .         .     99 

Other  forms  of  stairs  ......... 

CLASS   IV. 

Structures  containing  Double-Curved  Surfaces. 

PEOBLEM  XVII. — A  trumpet  bracket,  with  basin  and  niche     .         .         .103 
Theorem  III.  —  The  conic  section  whose  principal  vertex  and  point  of 
contact  with  a  known  tangent  are  given,  will  be  a  parabola,  ellipse, 
or  hyperbola,  according  as  the  given  vertex  bisects  the  subtangent, 
or  makes  its  greater  segment  within  or  without  the  curve         .         .106 

PEOBLEM  XVIII.  —  The  hooded  portal  ......       107 

PEOBLEM  XIX.  —  An  oblique  lunette  in  a  spherical  dome  .  .110 

Pendentives       ........  .114 

Spirals  .........  .115 

PEOBLEM  XX.  —  The  annular  and  radiant  groined  arch  .       120 


STEREOTOMY. 


STONE-CUTTING. 

FIRST    PRINCIPLES. 

1.  Stereotomy  is  that  application  of  Descriptive  G-eometry 
which,  comprehensively  defined,  treats  of  the  cutting  or  shap- 
ing of  forms,  whether  material  or  immaterial,  so  as  to  suit  cer- 
tain given  conditions. 

2.  Stereotomy,  thus  defined,  embraces,  either  by  etymology, 
or  established  usage,  the  following  subjects  :  — 

1°.  Shades  and  Shadows,  or  the  cutting  of  the  volume  of  space 
from  which  an  opaque  body  excludes  the  light,  by  any  given  sur- 
fac%,  on  which  the  shadow  of  the  body  is  thus  said  to  fall. 

2°.  Perspective,  or  the  cutting  of  the  cone,  of  which  the  ap- 
parent limit  of  a  given  body  is  the  base,  and  the  eye  the  ver- 
tex, by  any  given  plane,  whose  intersection  with  this  cone  is 
called  the  perspective  of  the  given  body. 

3°.  Dialing,  or  the  cutting  of  metal  plates  so  that  their  shad- 
ows upon  a  given  surface  shall  mark  the  hours  of  the  day. 

4°.  Cinematics,  or  the  shaping  of  mechanical  forms,  so  that 
by  their  mutual  action  they  shall  produce  certain  motions. 

5°.  Structural  articulations,  or  the  shaping  of  the  articula- 
tions of  wood  and  iron  framings  of  every  kind,  with  reference 
to  convenience  of  construction  and  use. 

6°.  Carpentry,  or  the  cutting  of  wooden  pieces,  so  that  when 
united  they  shall  form  a  self-supporting  whole. 

7°.  Stone-cutting,  or  the  cutting  of  stone  pieces  of  prescribed 
form,  from  the  rough  block,  so  that  when  combined  in  an  as- 
signed order,  they  shall  form  a  given  or  predetermined  whole. 

Of  these,  the  last  two  are  the  most  obviously  characteristic ; 
that  is,  most  clearly  illustrative  of  the  definition  (1). 

3.  Stone-cutting  as  a  science  embraces  three  distinct  parts:  — 
1°.  The  construction,  on  large  scales  in  practice,  of  the  pro- 
jections of  at  least  so  much  of  a  proposed  structure  as  will  per- 
mit— 

i 


2  STEREOTOMY. 

2°.  The  derivation  therefrom  of  the  directing  instruments, 
used  by  the  workman  as  guides  in  cutting  the  rough  block  to 
its  intended  form  by  the  chisel  and  mallet. 

3°.  The  rules  for  the  application  of  these  directing  instru- 
ments in  the  proper  order  and  manner. 

4.  The  first  two  of  the  three  parts  just  mentioned  consist 
of  operations  of  applied  descriptive  geometry.  The  number  of 
directing  instruments,  and  the  mode  of  their  application,  will 
depend  considerably  on  the  ingenuity  of  the  designer. 

5.  Practical  stone-cutting,  or  the  actual  formation  of  the 
finished  stone,  belongs  to  the  student,  only  so  far  as  it  may,  in 
the  absence  of  models,  serve  him  in  gaining  familiarity  with 
those  complex  masonry  forms  which  cannot  be  readily  imag- 
ined from  drawings  alone. 

In  such  modelling,  the  intended  pieces  would  be  wrought  in 
plaster,  by  the  aid  of  their  wooden,  or  paper  directors,  derived 
from  the  drawings. 

6.  Slopes  are  variously  expressed.     1°.  In  PL  L,  Fig.  1,  the 

Tk 
slope  H"k,  for  example,  may  be  expressed  by  the  ratio,         ;/,  = 

Tk         5 
the  tangent  of  the  angle  TH"k.    Here  ^7577  =  T '  rea^'  a  s^°Pe 

of  five  to  one. 

2°.  By  degrees.  Slopes  of  30°,  40°,  etc.,  make  these  angles 
with  the  horizontal  plane. 

3°.  A  batter  of  1  inch  to  1  foot,  etc.,  means  a  horizontal  de- 
parture of  one  inch  from  a  vertical  direction,  for  each  foot  of 
altitude. 

4°.  Nearly  level  slopes,  as  of  railways,  are  described  as  a  rise 
of  1  in  100,  etc.,  40  feet  to  the  mile,  etc. 

5°.  Once  more  ;  a  slope  of  45°  being  naturally  described  as 
that  of  1  to  1,  every  other  slope  may  be  described  by  naming 
its  horizontal  component  distance  first,  and  by  taking  its  least 
component  as  the  unit.  Thus  a  slope  of  4  horizontal,  to  1  ver- 
tical, may  be  described  as  a  slope  of  4  to  1.  But  one  of  1 
horizontal,  to  7  vertical,  for  instance,  as  a  slope  of  1  to  7 ;  or  a 
batter  of  1  in  7. 

The  nature  of  the  case,  or  a  reference  to  the  figure,  will 
show,  in  each  problem,  the  meaning  of  any  expression  of  slope 
that  may  be  used. 


STONE-CUTTING.  3 

Directing  instruments. 

7.  The  directing  instruments  (3)  used  in  stone-cutting  are 
of  three  kinds,  bevels,  templets,  and  patterns. 

Bevels,  as  the  common  steel  square,  give  the  relative  posi- 
tions of  required  lines,  or  surfaces  of  a  stone,  by  showing  the 
angles  between  them.  In  the  former  case  they  are  plane  bev- 
els ;  in  the  latter,  diedral  bevels. 

Templets  give  the  forms  of  required  edges,  or  other  distin- 
guishing lines  of  a  surface. 

Patterns  show  the  forms  of  plane  or  of  developable  surfaces. 
In  the  former  case,  they  may  be  made  of  any  stiff,  thin  mate- 
rial.    In  the  latter,  they  must  be  flexible. 

These  instruments  will  be  designated  by  numbers  in  the  sub- 
sequent problems,  and  in  every  case,  No.  1  will  be  a  straight- 
edge, and  No.  2  the  square. 

Notation. 

8.  For  the  sake  of  brevity,  the  horizontal  and  vertical  planes 
of  projection  will,  on  account  of  the  frequent  reference  which 
must  be  made  to  them,  be  denoted,  respectively,  as  the  planes 
H  and  V. 

The  usual  rules  for  inking  visible  and  invisible,  given  or 
auxiliary  lines  and  planes  (Des.  Geom.  45),  will  be  followed, 
unless  in  particular  cases  greater  clearness  may  result  from 
disregarding  them. 

When  important  lines  are  hidden  by  viewing  a  structure,  as 
usual,  vertically  downwards  from  above  it,  its  horizontal  pro- 
jection may  be  inked  as  if  the  object  were  seen  by  looking  at  it 
vertically  upward  from  below  it. 

The  greater  complexity  of  some  of  the  figures  will  make  it 
convenient  to  adopt  the  rule  of  distinguishing  invisible  lines 
of  the  structure  from  the  lines  of  construction,  by  dotting  the 
former  and  marking  the  latter  in  short  dashes  ;  a  distinction 
not  shown,  however,  on  the  plates  of  this  volume. 

9.  In  order  to  secure  a  brief,  yet  comprehensive  exhibition  of 
the  elements  of  stone-cutting,  that  is,  one  representing  every 
important  class  of  structures,  and  form  of  surface,  they  may 
be  classified  as  in  the  General  Table,  the  frontispiece. 


CLASS  I. 

Plane-Sided  Structures. 

Problem  I. 

To  form  plane  surfaces  of  stone,  making  any  given  angle  with 

each  other. 

This  fundamental  problem,  being  of  constant  occurrence,  is 
here  separately  explained,  in  order  to  avoid  repetition. 


Kg.  l- 


Fig.  2. 


10.  First.  Fig.  1.  represents  the  first  steps  in  forming  a 
plane  upon  a  wholly  unwrought  block.  Having  two  straight- 
edges, AB  and  CD,  of  equal  width,  ledges,  as  m  n  and  p  q,  are 
cut  on  opposite  edges  of  the  stone,  until  the  tops  of  the 
straight-edges  placed  on  them,  as  shown,  are  found  by  sight- 
ing, as  from  E,  to  be  in  the  same  plane. 

The  portion  of  rough  stone  between  m  n  and  p  q  is  then  cut 
away  until  found,  by  frequent  test  with  the  straight-edge,  ap- 
plied transversely,  as  at  FG,  to  be  wrought  down  to  the  plane 
of  m  n  and  p  q. 

Second.  Having  prepared  one  plane  surface,  as  shown  in 
Fig.  2,  a  second  plane  face,  perpendicular  to  the  former,  may 
be  formed,  as  shown,  by  cutting  two  or  more  channels  in  any 
direction  on  the  required  face,  until  one  arm,  BC,  of  a  square 
will  fit  any  of  them,  while  the  other  arm,  AB,  coincides  with 


STONE-CUTTING.  5 

the  given  face,  and  in  a  direction  perpendicular  to  the  common 
edge  of  the  two  surfaces.  The  intermediate  rough  stone  is  then 
cut  down  to  the  plane  of  these  channels  by  applying  the  straight- 
edge transversely  to  them.  For  any  other  than  a  right  angle, 
use  a  bevel  giving  such  angle. 

11.  The  principle  of  the  first  operation  is,  that  if  two  lines, 
m  n  and  p  q,  are  in  the  same  plane,  all  lines,  as  FG,  which  in- 
tersect them,  are  in  that  plane  also. 

That  of  the  second  operation  is,  that  if  two  planes  are  per- 
pendicular to  each  other,  any  line,  as  AB,  in  one  of  them,  and 
perpendicular  to  their  intersection,  will  be  perpendicular  to  all 
lines,  BC,  etc.,  drawn  through  its  foot  and  in  the  other  plane. 

Problem  II. 

A  sloping  wall  and  truncated  pyramidal  buttress. 

I.  The  Projections  (3) .  —  These,  PI.  I.,  Fig.  1,  are  made  partly 
from  given  linear  dimensions,  and  partly  from  given  slopes  of 
the  inclined  faces  of  the  buttress  and  wall. 

The  plan  and  front  elevation  can  be  made  wholly  from  data 
of  the  kinds  just  mentioned  ;  but  an  end  elevation  is  added, 
as  a  check  upon  errors,  and  as  showing  a  different  method  of 
operation. 

Let  the  wall  be  8  ft.  in  height,  and  3  ft.  6  in.  thick  at 
its  base,  and  with  a  slope  of  1  to  6  on  its  front.  Let  the  ver- 
tical height  of  the  buttress  at  c"  be  7  ft.  ;  the  slope  of  its  front 
1  to  5,  that  of  its  sides  1  to  4,  and  that  of  its  top  4^  to  1  (6). 
Then  — 

1°.  Construct  the  end  elevation  by  making  G"m  =  <t  of  G"E", 
and  A"C"  parallel  to  E"m  ;  also  E"N  =  10  ft.,  ON  =  2  ft.,  and 
Wh  parallel  to  OE"  ;'  and  C"L  =  4  ft.  6  in.,  LM  =  1  ft.,  and 
a"c"  parallel  to  MC",  having  made  c"e  1  ft.  below  C"E". 

These  operations  will  give  the  required  slopes  seen  in  the 
end  view. 

2°.  The  construction  of  the  plan  and  front  elevation  of  the  wall 
is  obvious  on  inspection,  all  the  heights  in  elevation  being  de- 
termined by  projecting  across  from  the  end  view,  and  all  the 
widths  in  plan,  by  transferring  the  horizontal  distances,  E"C", 
etc.,  from  the  end  view  to  any  convenient  line  of  reference,  as 
exh^  perpendicular  to  the  ground  line. 


6  STEREOTOMY. 

3°.  The  projections  of  the  buttress.  —  The  dimensions  of  its 
base  are,  JK  =  6  ft. ;  HI  =  4  ft. ;  and  A^U  =  2  ft.  6  in.  With 
these  draw  the  base,  whence,  by  means  of  the  end  view  and 
the  given  slopes,  all  its  lines  can  be  found. 

Thus,  ah  and  cd,  drawn  indefinitely  at  first,  are  at  distances, 
exbx  and  exd^  from  EF  respectively  equal  to  a"f  and  c"e. 

The  slope  of  the  sides  being  1  to  4,  draw  ij  parallel  to  HJ, 
and  at  a  distance,  gi,  from  it,  equal  to  one  fourth  of  G''E". 
Then  Qj  and  ij,  being  in  the  plane  of  the  top  of  the  wall,  Jj  is 
the  intersection  of  the  face  of  the  wall  with  the  left  side  of  the 
buttress ;  and  c,  its  intersection  with  dxd,  is  a  back  upper  cor- 
ner of  the  buttress.  Drawing  the  horizontal  a"f,  and  making 
elnl,=fn",  draw  nxn  parallel  to  AB,  and  from  n,  draw  na  par- 
allel to  JH,  and  na  will  meet  ah  at  a.  Then  draw  ac  and  aH, 
which  will  complete  the  left  side  of  the  buttress.  Its  right  side 
can  be  laid  off  from  the  axis  of  symmetry  UV. 

The  vertical  projection  of  the  buttress  is  made  by  simply 
projecting  the  points  of  the  plan  up  to  the  traces  H'K',  a'b', 
and  c'd'  of  the  horizontal  planes  in  which  they  lie,  and  then 
joining  the  points  as  shown. 

In  constructions  like  that  of  Jj,  the  pairs  of  parallels,  A  J 
and  HJ  ;  C/  and  ij,  which  determine  the  required  line,  should 
be  as  far  apart  as  possible. 

II.  The  Directing  Instruments.  —  These  are,  besides  Nos.  1 
and  2  (7),  No.  3,  a  pattern  of  the  base  of  the  upper  stone  of 
the  buttress,  and  seen  in  its  true  size  in  plan  :  No.  4,  a  bevel 
containing  the  diedral  angle,  shown  on  the  end  elevation,  be- 
tween the  base  and  the  back  of  the  buttress :  No.  5,  the  like 
angle  between  the  base  and  front  of  any  buttress  stone  :  No.  6, 
a  bevel  giving  the  angle  between  the  base  and  either  of  the 
sides  of  the  buttress. 

No.  6  is  found  by  revolving  any  line  of  greatest  declivity,  as 
R/i,  perpendicular  to  JH,  until  parallel  to  V,  as  at  RA"  — Wh'n ; 
when  it  will  show  the  true  slope  of  the  side  of  the  buttress. 
No.  9  gives  the  angle  between  the  top  and  back  of  the  but- 
tress. 

If  patterns  are  used  as  checks  upon  the  operation  of  the  bev- 
els, they  may  be  found  as  follows. 

No.  7  is  a  pattern  of  the  top  of  the  buttress,  and  shown  in 
its  true  form  by  revolution  about  cd  —  c'd',  till  parallel  to  V. 
This  is  done  by  making  c'"V  =  a"c"  and  drawing  a'"b"r  paral- 


STONE-CUTTING.  7 

lei  to  a'b'  and  limited  by  the  vertical  projections,  oV"  and  b'b'", 
perpendicular  to  c'd',  of  the  arcs  described  by  ad  and  bb'  in  re- 
volving about  cd —  c'd'. 

No.  8  is  a  pattern  of  the  right  side  of  the  upper  abutment 
stone,  and  is  shown  in  its  true  form  by  revolving  it  about  PQ 
—  P'Q'  till  horizontal.  Then  d",  the  revolved  position  of  dd', 
may  be  found  by  describing  an  arc  with  P  as  a  centre  and  a 
radius  equal  to  d'P'",  shown  by  the  construction  to  be  the  true 
length  of  Pd  —  P'd',  and  noting  where  it  intersects  dd"  per- 
pendicular to  the  axis  PQ.  The  point  b"  being  similarly  found, 
PQb"d"  is  the  required  pattern.  These  patterns  would  con- 
veniently be  open  wooden  frames. 

III.  The  Application.  —  We  will  here  illustrate  this  topic  by 
the  manner  of  working  the  top  stone  of  the  buttress.  This, 
being  the  most  irregular  stone  in  the  problem,  the  full  explana- 
tion of  the  manner  of  forming  it  will  enable  the  student  to  de- 
vise means  for  working  any  of  the  others. 

Having  selected  a  rough  block  in  which  the  finished  stone 
might  be  inscribed,  first  bring  the  intended  base  of  the  stone  to 
a  plane  by  Prob.  I.  Next  scribe  the  edges  of  the  base  by  pat- 
tern No.  3. 

Having  thus  the  bottom  edges  of  all  the  lateral  faces  of  the 
stone,  these  may  be  wrought  from  the  base,  and  in  their  true 
relative  position,  by  the  bevels  Nos.  4,  5,  and  6.  By  marking 
the  distances  sc"  and  ra"  on  bevels  Nos.  4  and  5  respectively, 
the  edges  ab  —  a'b'  and  cd  —  c'd'  will  be  determined  on  the 
stone,  which  can  then  be  finished  by  the  use  of  No.  9  and  the 
straight  edge. 

For  increased  accuracy,  patterns  Nos.  7  and  8,  and  others 
similarly  found  for  the  front  and  rear  faces  of  the  stone,  can  be 
used  in  determining  the  edges  of  the  lateral  faces  more  posi- 
tively than  by  the  natural  intersection  of  these  faces  as  found 
by  the  bevels  and  straight  edge. 

Examples.  — 1°.  Construct  patterns  10  and  11,  of  the  front  and  rear  faces  of 
the  top  buttress  stone. 

Ex.  2°.  Consmict  a  bevel,  No.  12,  giving  the  angle  between  the  front,  and  an 
adjacent  lateral  face. 

Ex.  3°.  Show  the  construction  and  use  of  the  guides  necessary  in  working  the 
stone  below  the  top  one  of  the  buttress. 

Ex.  4°.  Do.  for  the  top  stone,  supposing  ijg  top  surfaces  to  be  a  pyramid  hav- 
ing its  vertex  at  vv'. 

Ex.  5°.  Construct,  on  a  larger  scale  than  in  Fig.  5,  a  plan  and  front  elevation 


b  STEREOTOMY. 

of  the  wall  whose  end  elevation  is  there  shown,  with  an  isometrical  figure  of  the 
stone  NFEDCg1.  [The  inclined  surfaces,  aspq,  help  to  prevent  sliding  in  the  direc- 
tion of  the  arrow,  from  the  pressure  of  materials  at  the  back,  AN,  of  the  wall.] 

Problem  III. 

The  recessed  flat  arch,  or  plate-band. 

12.  An  arch  is  an  assemblage  of  blocks,  mutually  support- 
ing, by  means  of  radiating  joints  between  them,  and  side  sup- 
ports to  confine  them  laterally.  When  the  arched  surface, 
usually  cylindrical,  is  plane,  the  structure  is  called  a  Plate- 
band. 

I.  The  Projections.  —  PL  I.,  Figs.  2,  3,  and  4.  The  con- 
struction of  these  will,  after  a  brief  description,  be  sufficiently 
obvious  on  inspection  of  the  figures,  since,  on  all  of  them,  the 
same  letter  indicates  the  same  point. 

Tt  Jj — T/T"J7U/,  Fig.  2,  is  the  rectangular  door-way,  or 
opening  through  a  wall.  The  door,  not  shown,  folds  against 
the  vertical  plane  surface,  U'V'J'Q'T"S"S'T',  which  is  in  the 
plane  qs.  The  recess  which  contains  the  door  when  shut,  is 
bounded  by  the  surfaces  just  pointed  out,  and  by  the  three 
surfaces,  Qq  —  V'Q';  Q^Ss—  Q'S" ;  and  Ss  —  S'S",  all  of 
which  are  perpendicular  to  the  plane  V- 

The  surfaces  PQ  —  V'W'P'Q',  PQRS  —  PQ'R"S",  and 
E,S  —  R/S'R/'S",  collectively,  form  the  convergent  sides,  or 
jambs  of  the  door- way. 

The  joints,  AT)',  etc.,  divide  T"J'  into  equal  parts,  and 
radiate  from  a  point  O,  found  by  making  OT"J'  an  equilateral 
triangle.  The  lapping  of  the  end  stones,  as  Y,  over  the  upper 
jamb  stones,  as  at  G'F',  is  designed  to  give  greater  security. 

Fig.  3  is  an  oblique  projection  of  the  stone  A'D'E'F',  as  seen 
in  looking  at  it  obliquely  upward  so  as  to  see  its  front,  right- 
hand,  and  under  surfaces.*  The  identity  of  its  several  sur- 
faces with  the  same  ones  as  seen  in  Fig.  2,  is  shown  by  the 
lettering. 

Fig.  4  is  an  isometrical  drawing  of  the  upper  jamb-stone 
G'X'N',  showing  its  front,  left-hand,  and  under  surfaces. 

II.  The  Directing  Instruments.  —  Of  these  the  only  ones  re- 
quiring any  further  graphical  operations  for  their  construction 

*  See  my  Elementary  Projection  Drawing  :  Div.  IV.  Chap.  V.,  for  an 
explanation  of  this  species  of  projection. 


STONE-CUTTING.  9 

are  the  patterns  of  the  radiating  joints.  One  of  these  is  shown 
at  l'p'pnf"!f"k'"k",  by  revolving  the  joint,  vertically  projected 
in  f'p',  about  that  line,  considered  as  in  the  plane  RX,  as  an 
axis ;  whence  p'p"=Rr ;  k'k"=cu ;  k'k"f=cA,  etc. 

The  true  form  of  the  joint  on  J'G'  may  be  similarly  found. 

III.  The  Application.  —  Let  it  be  required  to  work  the  stone 
A'D'F',  Figs.  2  and  3.  There  is  no  one  precise  order  of  oper- 
ations which  is  indispensable  ;  but  we  should  naturally  work 
first,  either  the  top  surface,  EeD<i,  as  the  simplest,  or  the  back 
deaj,  or  joint  on  A'D',  as  the  largest  and  most  important. 

Supposing  then  that  we  have  a  block  in  which  the  stone  can 
be  inscribed,  as  shown  at  fhmo;  bring  the  intended  joint  sur- 
face on  A'D'  to  an  indefinite  plane  by  Prob.  I.,  and  by  the 
pattern,  No.  3,  mark  its  edges.  Next,  the  back  and  front  can 
both  be  wrought  square  with  this  surface  by  No.  2,  and  their 
edges  marked  by  their  patterns  A'D'E'F'G'J'  (No.  6),  and 
C'D'E'F'G'H'  (No.  7).  The  top  and  bottom  surfaces  can 
then  be  wrought,  like  all  the  lateral  surfaces,  either  square 
with  the  back,  by  No.  2 ;  or,  from  the  joint  surface  A'D'  by 
the  bevels  Nos.  4  and  5. 

The  surfaces  on  E'F'  and  G'F'  are  square  with  each  other 
and  with  the  front  and  back.  A  pattern,  No.  8  (not  shown), 
of  the  joint  on  J'G',  and  a  bevel,  No.  9,  set  to  the  angle  QPX, 
between  the  vertical  and  jamb  surfaces,  will  suffice  for  the  ac- 
curate completion  of  the  stone. 

13.  The  stones  A'D'F'  and  G'F'N'  cannot  be  more  clearly 
shown  than  in  Figs.  3  and  4,  in  which  all  their  most  irregular 
portions  are  made  visible.  But  the  student  cannot  here  be- 
come too  familiar  with  the  methods  of  representation  there 
given,  on  account  of  their  special  usefulness  in  the  drawing  of 
complicated  masonry.  Hence  the  nature  of  a.  part  of  the  fol- 
lowing examples. 

Ex.  1°  Make  an  isometric  drawing  of  the  stone  Y,  as  inscribed  in  a  prism 
containing  its  top  and  bottom  faces. 

Ex.  2°.  Make  the  same  drawing  of  Y  as  inscribed  in  the  prism  fhmo. 

Ex.  3°.  Make  an  oblique  projection  of  the  upper  left-hand  jamb-stone. 

Ex.  4°.  Draw  the  instruments,  and  describe  their  use,  for  working  the  upper 
jamb  stones  as  G'F'N'. 

Ex.  5°.  Devise  and  describe  any  other  convenient  manner  of  working  the  stone 
Y,  than  that  given  in  the  text. 

Ex.  6°.  Make  the  top  of  the  door-way  semi-octagonal,  by  sloping  the  under 
sides  of  Y  and  A'D'G'  at  45°  to  the  vertical  sides,  on  Tt  and  J/,  of  the  opening. 


10  STEREOTOMY. 


Plane  Sided   Wing-Walls. 

14.  Wing-walls,  are  those  which  flank  the  approach  to  any 
passage-way,  or  area,  so  as  to  prevent  obstructing  it. 

15.  PI.  L,  Figs.  6-9,  illustrate  in  outline  the  leading  forms 
of  plane  sided  wing-walls ;  rectilinear,  prismatic,  and  pyram- 
idal. 

Fig.  6,  a  rectilinear  wing- wall,  AB  —  A'B',  is  such  as  might 
be  used  to  retain  a  roadway  ascending  behind  it,  and  bending, 
as  shown  by  the  arrows,  to  cross  a  bridge,  one  of  whose  abut- 
ments is  CC.  Such  a  construction  is  often  seen  where  a  rail- 
way or  canal  is  crossed  at  right  angles  by  a  road  which,  on  one 
or  both  sides  of  the  crossing,  is  parallel  to  the  railway  or  canal. 

Fig.  7  represents  wing-walls,  as  abc  —  b'a'c',  which,  with  the 
arch-wall,  cd  —  c'd',  form  three  sides  of  a  vertical  hollow  quad- 
rangular prism,  in  which  the  tops  of  the  wing- walls  are  trun- 
cated by  an  oblique  plane,  a'c'd'g'.  Had  there  been  a  slope  or 
batter  to  the  arch  and  wing-walls,  the  inner  surfaces,  collec- 
tively, would  have  formed  part  of  a  vertical  quadrangular  pyra- 
mid. Either  construction  would  serve  in  case  of  a  railway 
tunnel  occupying  the  line  of  a  city  street. 

Fig.  8  represents  wing-walls,  which,  with  the  connecting 
arch-wall,  DF,  form  half  of  a  vertical  hollow  hexagonal  prism, 
wholly  truncated  by  an  oblique  plane,  AEG  —  A'E'G'  ;  and 
flanking  the  inlet  to  a  pipe  culvert,  mn  —  m'n',  by  which  the 
water  of  some  gulley  or  small  ravine  may  be  led  through  an 
embankment. 

Fig.  9  differs  from  Fig.  8,  in  that  the  top  of  the  arch-wall, 
dstg  —  d'g',  is  horizontal,  and  the  inner  faces  of  the  walls  form 
a  part  of  a  vertical  inverted  pyramid. 

16.  A  coping  is  a  layer  of  thin  stones,  laid  on  the  top  of  a 
wall,  and  made  broader  than  the  thickness  of  the  wall,  which 
they  cover,  for  the  purpose  of  sheltering  the  joints  from  the 
weather,  while  their  adaptation  to  this  end  and  the  effect  of 
their  shadows,  add  to  the  beauty  of  the  whole. 

Had  the  arch-wall,  Fig.  9,  been  truncated  by  the  horizontal 
plane  P'Q',  it  might  have  received  a  coping,  whose  front  edge 
would  have  been,  in  plan,  at  pq,  formed  by  projecting  r'  at  r. 
Lines  between,  and  parallel  to  be  and  fk,  would  represent  the 
front  of  the  copings  of  the  wing- walls. 


STONE-CUTTING.  11 

Problems  I.  —  III.  will  enable  the  student  to  work  the  fol- 
lowing — 

Examples.  —  1°.  Draw  PI.  I.,  Fig.  7,  to  scale  from  measurements,  with  a  section 
on  the  plane  nNn',  and  describe  the  cutting  of  a  stone  at  the  corner  dd',  which,  to 
break  joints  with  courses  below,  shall  extend  from  kf  in  the  arch- wall,  cd,  to  pj 
in  the  wing-wall,  ee'. 

Ex.  2°.  Draw  PI.  I.,  Fig.  8,  in  like  manner,  making  the  vertical  joint  surfaces, 
perpendicular  to  FD  and  FG,  and  describe,  with  an  isometrical  or  oblique  projec- 
tion, the  working  of  the  top  stones  at  one  of  the  angles  D  or  F. 

Ex.  3°.  In  Ex.  2°,  let  there  be  a  coping,  and  let  the  top  be  truncated  as  at  P'Q', 
Fig.  9. 

Ex.  4°.  Treat  PI.  I,  Fig.  9,  as  described  in  Ex.  2°. 

Ex.  5°.  In  Ex.  4,  let  there  be  a  coping.  Also  add  square  piers  at  the  foot  of 
the  wing-walls,  each  side  of  which  in  plan  shall  be  equal  to  ac,  and  whose 
height  shall  be  a'a" . 


CLASS  II. 

Structures  containing  Developable  Surfaces. 

17.  Walls  having  either  cylindrical  or  conical  faces,  will  be 
treated  incidentally,  in  connection  with  other  constructions  of 
which  they  form  a  part. 

Arches  (12)  include  as  particular  cases,  portals  or  arched 
openings  through  walls  ;  and  combined,  or  groined,  and  clois- 
tered arches.  Vaults  include,  more  comprehensively,  domes, 
and  other  over-arched  areas. 

Only  cylindrical  arches  will  now. be  considered,  and  under 
the  heads  of,  — 

1°.  Definitions.  III0.  Preliminary  Constructions. 

11°.  Classification.       IV°.  Practical  Illustrations. 

1°.  —  Arches  —  Definitions. 

18.  Arch  Masses.  —  The  supporting  walls  of  an  arch  are 
called  its  abutments  when  backed  by  earth,  etc.,  and  piers, 
when  exposed  on  all  sides.  The  superincumbent  masonry, 
supported  by  the  arch,  is  called  the  spandril,  or  backing. 

Each  row  of  blocks  extending  through  the  length  of  the 
arch  is  a  course.  Each  block  in  the  course  is  an  arch-stone, 
or  voussoir.  The  extreme  voussoirs  of  each  course  are  the 
quoins,  or  ring-stones.    The  middle  ring-stone  is  the  key-stone. 

19.  Arch  /Surfaces.  —  The  end  surfaces  of  an  arch  are  its 
faces,  a'd'  —  aQDd,  PI.  III.,  Fig.  27.  (In  certain  cases,  where 
the  elevation  is  more  important,  or  naturally  made  first,  the 
accented  letters  are  attached  to  the  plan  (8)  ).  The  inner 
cylindrical  surface,  ABD,  is  the  intrados.  The  outer  one, 
whether  cylindrical,  Fig.  29,  or  plane-sided,  Fig.  27,  is  the 
extrados.  The  top  of  the  abutment,  if  level,  is  called  the 
springing  plane  ;  if  inclined  radially,  the  skew-back. 

The  longer  sides,  mm'rr',  Fig.  26,  of  a  voussoir,  are  its  beds, 
and  are  continuous  from  stone  to  stone.  Its  ends,  are  its  heads, 
and  these  terminate,  or  break,  in  the  coursing  surfaces.  Its  in- 
ner side  is  its  soffit ;  the  outer,  its  back. 


STONE-CUTTING.  13 

20.  Arch  Lines.  —  The  intersections  of  the  faces  with  the 
intrados  and  extrados  are  the  face  lines  of  the  arch,  and  of 
those  surfaces.  The  intersections  of  the  springing  plane,  or 
the  skew-back,  with  the  same  surfaces,  are  the  springing  lines. 
The  axis  joins  the  centres  of  the  face  lines,  and  is  parallel  to 
the  springing  lines  in  an  arch  of  uniform  cross  section,  and  is 
also  straight  in  a  cylindrical,  and  a  conical  arch.  The  edges, 
parallel  to  the  a±is,  of  the  courses  are  the  coursing  joints ; 
the  transverse  edges  of  the  voussoirs  are  the  heading  joints  ; 
the  edges  lying  in  the  thickness  of  the  arch  are  the  radial 
joints,  and  the  joints  in  the  face  are  the  face  joints.  The 
span  is  the  perpendicular  distance  between  the  springing  lines. 
The  rise  is  the  greatest  height  of  the  intrados  above  the  span. 

21.  Arch  Points.  —  The  highest  point  of  any  right  section  of 
an  arch  is  its  crown.  The  intersection  of  the  axis  with  the 
face  is  the  face  centre.  When  the  face  joints  radiate  from  any 
other  point  than  the  face  centre,  such  point  is  called  a,  focus. 

11°.  —  Arches  —  Classification. 

22.  Arches  may  be  classified  according  to  either,  1°,  the  form 
of  the  intrados  ;  2°,  the  relation  of  the  axis  to  other  parts  ;  or 
3°,  the  forms  of  the  face  lines. 

The  two  former  are  the  most  important  grounds  of  division, 
since  they  give  rise  to  more  radical  differences  of  design.  The 
latter  occasions  only  the  many  minor  modifications  of  form. 

23.  Arches,  divided  according  to  the  forms  of  their  intra- 
dos, are  plane,  developable,  ivarped,  or  double-curved. 

Plane  arches  have  been  illustrated  in  Prob.  III. 

Developable  arches  are  those  in  which  the  intrados  is  near- 
ly always  cylindrical. 

Warped  and  double-curved  arches  will  be  treated  in  con- 
nection with  other  problems  involving  like  surfaces. 

24.  Arches,  divided  according  to  the  direction  of  the  axis, 
relative  to  other  parts,  are  right,  oblique,  descending,  and  ram- 
pant. 

A  right  arch  is  one  in  which  the  axis  is  horizontal,  and  the 
planes  of  the  faces  at  least  so  nearly  perpendicular  to  it  that 
the  coursing  joints  can  all  be  parallel  to  the  axis.  The  face 
may  be  oblique  to  H,  when  it  has  a  batter ;  or  oblique  to  a 
vertical  plane  through  the  axis,  when  it  is  a  skew-face.  It  may 
be  oblique  to  both  of  these  planes  at  once. 


14  STEREOTOMY. 

25.  The  oblique  arch,  properly  so  called,  is  one  in  which  the 
coursing  joints,  in  order  to  be  nearly  or  quite  perpendicular 
to  the  face,  are  not  parallel  to  the  axis,  the  latter  being  hori- 
zontal, and  oblique  to  the  horizontal  lines  of  the  face. 

26.  A  descending  arch  is  one  whose  axis  is  inclined  to  a 
horizontal  plane. 

A  rampant  arch  is  one  in  which  one  springing  line  is  higher 
than  the  other.  The  descending  arch  is  also  "sometimes  called 
rampant.  The  two  cases  may  then  be  distinguished  as  longi- 
tudinally rampant,  when  one  end  of  the  axis  is  higher  than  the 
other,  and  transversely  rampant,  when  one  springing  line  is 
higher  than  the  other. 

27.  Arches,  divided  according  to  the  form  of  the  face  line, 
may  be  circular,  elliptic,  parabolic,  etc.,  according  to  the  form 
of  the  right  section  of  the  intrados.  They  are  also  depressed, 
PL  II.,  Figs.  13,  15,  or  super-elevated,  Fig.  14,  according  as 
the  rise  (20)  is  less  or  more  than  half  of  the  span.  They  are 
also  pointed,  if,  as  in  Fig.  14,  the  right  section  is  composed  of 
intersecting  arcs. 

When  the  right  section  of  an  arch  is  a  semicircle,  semi- 
ellipse,  etc.,  the  arch  is  said  to  be  full-centred.  When  this 
curve  is  less  than  a  half  one,  the  arch  is  called  segmental. 

28.  When  the  right  section  is  compound,  having  three,  five, 
or  many  centres,  the  arch  is  said  to  be  three-centred,  etc.,  or 
poli/central,  PL  III.,  Figs.  11,  21. 

111°.  —  Arches  —  Preliminary  Constructions. 

§  Conic  Sections. 

1°.  —  To  construct  a  circle  by  points,  having  given  its 
radius. 

29.  Let  Oe,  PL  II.,  Fig.  10,  be  the  radius  of  the  required 
circle,  and  cd  =  20e  the  diameter  perpendicular  to  Oe.  Draw 
ep  and  cp,  which,  with  Oe  and  Oe,  will  form  a  square,  and 
hence  be  tangent  at  c  and  e.  Divide  Oe  and  ep  into  the  same 
number  of  equal  parts,  numbered  as  in  the  figure.  Then  lines 
radiating  from  c  and  d,  through  the  like  points  on  ep  and  Oe, 
respectively,  will  meet  at  points  v,  r,  etc.,  of  the  circle  whose 
radius  is  Oc.  For  Ofd  and  cgp  are  equal  right  triangles,  and 
the  angle  Odf  is  moreover  common  to  the  triangles  Ofd  and 


STONE-CUTTING.  15 

One.  Then  by  subtraction,  the  angles  ned  and  Ofd  are  equal, 
which  makes  the  triangles  Ofd  and  den,  similar,  and  hence  den 
right  angled  at  n.  Hence  n  is  a  point  of  the  semicircle  whose 
diameter  is  cd. 

2°.  — To  construct  an  arc  of  a  circle  by  points,  knowing 
its  chord  and  versed-sine  or  rise. 

30.  Let  2cb,  PI.  II.,  Fig.  15,  be  the  given  chord,  and  ab  the 
given  rise  of  the  required  arc.  Draw  ac,  and  ee  perpendicular 
to  it,  and  limited  by  the  tangent  ae  at  a ;  also  en  parallel  to 
ab.  Then  divide  cb,  ae,  and  en,  each  into  the  same  number 
of  equal  parts,  and  number  them  as  in  the  figure.  Then  lines 
joining  like  points  on  cb  and  ae,  will  meet  those  radiating  from 
a  to  the  points  on  en  in  points  of  the  required  arc. 

Parallels  to  Oe  and  Oe,  Fig.  10,  from  any  point  of  the  arc 
ee,  will  show  the  correctness  of  this  construction,  and  of  points 
as  k  on  the  arc  ae,  Fig.  15,  produced. 

Making  cc'  radial  and  equal  to  aa'  we  may  likewise,  as  seen 
in  the  figure,  construct  the  concentric  arc  a'cr  of  the  extrados 
of  a  segmental  arch  (27). 

31.  The  radius  of  the  segmental  arch,  given  by  its  span  and 
rise,  may  be  desired,  and  may  be  found  thus.  PL  II.,  Fig. 
15. 

Let  C,  intersection  of  abC  and  rC,  the  bisecting  perpendicu- 
lar to  ae,  be  the  centre  of  the  arc  ac.  Then  let  R  =■  Ca  ;  s  =■ 
be  and  v  =  ab.     Now  ab  :  ae  : :  ar  :  aC. 

Whence  aO  =  ac  X,  ar . 
ab 


But  ar  —  h  ae  ;   hence 
aC  = 


ac2 s2  -\-  v2 

2ab~~       2v     ' 


3°.  —  To  construct  an  ellipse  by  points  on  given  axes.   Also, 
normals  to  it. 

32.  Let  AB  and  CD  be  the  given  axes,  PI.  II.,  Fig.  16. 
Then  as  the  projection  of  a  circle  seen  obliquely  is  an  ellipse 
(Des.  Geom.,  Theor.  VI.),  the  curve  may  be  found  as  at  a,  b, 
c,  as  in  Fig.  10,  as  is  evident  by  comparing  Figs.  10  and  16. 

In  an  elliptic  arch  the  face  joints  should  be  normal  to  the 
elliptic  face  line  of  the  intrados. 

33.  Construction  of  normal  Face  Joints.     First  Method.  — 


16  STEREOTOMY 

Let  N  be  a  point  at  which  such  a  joint  is  to  be  drawn.  With 
C  or  D  as  a  centre,  and  a  radius  equal  to  Ah,  describe  arcs 
intersecting  AB  at  F  and  F,  which  are  the  foci  of  the  ellipse. 
Then  the  normal  Nm,  at  N,  bisects  the  angle  FNF. 

34.  Second  Method.  —  The  normal  at  any  point  being 
perpendicular  to  the  tangent  at  the  same  point,  to  draw  the 
normal  at  T,  for  example ;  describe  an  arc  as  At,  with  cen- 
tre h,  and  radius  Ah,  and  produce  dT,  perpendicular  to  AB,  to 
meet  this  arc  at  t.  This  arc  may  represent  the  circle,  which, 
when  revolved  about  AB  till  oblique  to  the  paper,  is  projected 
in  the  given  ellipse.  Then  draw  the  tangent  tK  to  the  arc, 
and  as  K,  being  in  the  axis  AB,  remains  fixed,  KT  is  the  tan- 
gent to  the  ellipse  at  T.  Then  Tk,  perpendicular  to  KT,  at 
T,  is  the  required  normal  at  T. 

4°. —  To  construct  the  arc  of  a  parabola;  on  a  given  seg- 
ment of  the  axis,  and  the  chord  which  is  perpendicular 
to  the  axis.    Also,  normals  to  it. 

35.  On  comparing  Figs.  10, 16,  and  20,  PI.  II.,  and  knowing 
from  geometry  (Des.  Geom.,  Arts.  206-209),  that  the  several 
conic  sections  have  certain  general  properties  in  common,  their 
construction  may  be  put  in  general  terms  applicable  to  all  cases, 
thus.  Points  of  any  conic  section  are  found  at  the  intersection 
of  sets  of  lines  which  radiate  from  the  two  vertices  of  an,  axis 
of  the  curve ;  those  from  one  vertex  radiating  to  points  of  equal 
division  on  the  common  perpendicular  to  the  chord  and  to 
the  tangent  at  that  vertex ;  and  those  from  the  other  vertex,  to 
like  points  on  the  semi-chord,  in  the  manner  shown  in  the  fig- 
ures. Thus,  PL  II.,  Fig.  20,  let  AU  be  a  semi-chord,  and  BU 
a  segment  of  the  axis  of  a  required  parabola.  Draw  the  tan- 
gent BV,  limited  by  AV,  parallel  to  BU.  Divide  AV  and 
AU  into  the  same  number  of  equal  parts,  and  number  like 
points,  reckoned  from  A,  with  like  numbers.  Then  radials 
from  B  to  the  points  on  AV,  will  meet  radials  from  the  oppo- 
site vertex  (B')  of  the  axis  BU  to  like  points  on  AU,  in  points, 
k,  c,  N,  etc.,  of  the  parabola.  But  note  that  as  the  axis  is  of 
infinite  length,  the  radials  from  its  infinitely  distant  extremity, 
B',  will  be  parallel  to  BU  as  shown  at  Nm,  etc. 

The  figure  further  shows  a  depressed  gothic  arch,  each  half 
composed  of  parabolic  arcs,  as  AB  and  DE. 

36.  Construction  of  the  Normal  Joints.  —  Let  such  a  joint 


STONE-CUTTING.  17 

be  constructed  at  N.  Draw  NN'  perpendicular  to  the  axis 
BU,  make  BT  =  BN',  and  TN  will  be  the  tangent  at  N  by 
the  property  that  the  subtangent,  N'T,  is  bisected  by  the 
vertex,  B,  of  the  curve.  Then  Nw,  the  normal,  is  perpendicular 
to  the  tangent  NT. 

In  the  parabola,  the  sub-normal  N'w  is  constant ;  hence,  to 
construct  any  other  normal,  as  at  e,  we  have  only  to  draw  ee' 
perpendicular  to  BU,  make  e'h'  =  N'w,  and  h'e  will  be  the  nor- 
mal at  e. 

37.  PI.  II.,  Fig.  13,  represents  a  depressed  gothic  arch 
formed  of  circular  arcs,  in  imitation  of  Fig.  20.  The  con- 
struction (Draft.  Insts.  &  Oper.  p.  91)  may  be  obvious  on  in- 
spection. The  joints  in  each  arc  radiate  to  the  centre  of  that 
arc.  Fig.  14  represents  a  pointed  or  lancet  gothic  arch,  where 
the  radii,  as  ca  and  cb,  are  greater  than  the  span,  de. 

5.  —To  construct  an  arc  of  a  hyperbola,  on  a  given  chord, 
and  segment  of  the  axis  perpendicular  to  the  chord. 

38.  This  form  of  the  conic  section  is  seldom  needed  by  the 
mason.  Yet  to  complete  the  series  of  similar  constructions  it 
is  here  given.  Let  act,',  PI.  II.,  Fig.  18,  be  the  chord,  and  Ve 
the  segment  of  the  axis.  Make  a't'  and  at  equal  and  parallel 
to  Ve,  and  divide  at  and  ae  similarly,  as  in  the  previous  ex- 
amples. Then,  WV  being  that  axis  of  the  hyperbola  which 
is  perpendicular  to  the  given  chord  aa\  radials  from  V  to  the 
points  on  at  will  meet  those  from  W,  to  the  like  numbered 
points  on  ae,  in  points  as  5,  c,  d,  of  the  required  hyperbola. 

39.  This  construction  is  essentially  the  perspective  of  that 
of  Fig.  10  on  a  plane  so  placed  as  to  cut  the  visual  cone  whose 
base  is  the  circle,  in  a  hyperbola  ;  while  that  of  Fig.  20  is 
similarly  the  perspective  of  Fig.  10  upon  a  plane  cutting  the 
visual  cone  whose  base  is  the  circle,  in  a  parabola. 

40.  In  Figs.  16,  18,  and  20,  we  may  substitute  any  chord 
and  its  conjugate  diameter  (that  is,  the  one  which  bisects  the 
given  chord  and  its  parallel  tangent)  for  the  axis  and  a  chord 
perpendicular  to  it ;  since  such  constructions  would  merely  be 
the  projections,  or  perspectives  of  Fig.  10  on  planes  not  paral- 
lel to  either  of  the  given  diameters,  Oe  and  Oe. 

2 


18  STEREOTOMY. 

§§    Polycentral  Arch- Curves. 

41.  The  practical  convenience  of  circular,  above  that  of 
other  curved  work,  with  other  reasons  which  will  appear,  has 
led  to  the  frequent  adoption  of  artificial,  or  compound  curves, 
composed  of  circular  arcs  of  different  radii,  in  place  of  the  true 
conic  sections ;  especially  in  case  of  separate  elliptic  arches, 
i.  e.  when  not  combined  as  in  groined  arches.  And  as  there 
will  be  found  to  be  a  choice  in  the  relative  lengths  of  these 
radii,  they  should  not  be  chosen  arbitrarily,  but  so  as  to  give  the 
best  proportions  to  the  intended  oval,  as  will  next  be  shown. 

Three-Centred  Ovals. 

1°.  To  construct  the  general  case  of  the  semi-oval  of  three 
centres. 

42.  In  PL  II.,  Fig.  17,  let  ab  and  bg  be  the  given  semi-axes 
of  the  proposed  curve.  Assume  ac  at  pleasure,  but  less  than 
bg;  make  ge  =  ac,  and  draw  ce  ;  bisect  ce  by  the  perpendicular 
to  it,  fd,  meeting  gb  produced  at  d.  Then  will  c,  d,  and  a 
point  <?1?  taken  on  ah  produced  so  that  bcx  =  be,  be  the  three 
centres  of  the  required  oval. 

43.  From  this  beginning,  the  following  investigation  leads  to 
useful  special  cases.     Let 

ab  =  a  ;  bg  =  b  ;  ac  =  r  ;  dg  =  R. 
Then  from  the  triangle  bed  we  have, 

(cdy  =  (bdy  +  (bcy. 

That  is  (R  —  r)2  =  (R  —  5)2  +  (a—  r)2  (X) 

whence  ^ a2  -\-  V  —  2ar  ,Q. 

K  -      2(6  —  r)  W 

equations  which  obviously  allow  an  infinite  number  of  solutions 
for  the  same  given  axes  ;  as  the  construction  shows,  where 
ac,  =  r,  was  assumed  at  pleasure,  only  less  than  bg  ;  and  thence 
R  was  found. 

2°.  First   special  case.     The  semi-oval  of  three  centres, 
■when  the  lesser  arc  is  60°. 

44.  Let  the  arc  am,  PI.  II.,  Fig.  11,  =  60°  ;  whence  bed  = 
cdc!=  60°,  and  the  arc  mn=.  60°.  Now  put  r  =  a  —  x,  where 
bc=  x,  then  R  =  md  =  ac'  =  a  -J-  x. 


STONE-CUTTING.  19 

Substituting  these  values  of  R  and  r  in  (1)  Art.  43,  we  find 
ar  —  (a  —  o)x  ■==   - — •£-!— 

whence,  neglecting  the  negative  value  of  x, 

a  —  b    -    a  —  b      .— 
*  =  -g—  +  — g—  V  3,  (4) 

which  is  constructed  as  follows  : 

Take  bf  =  a  —  5,  bh  —  — ^ — ,  and  on  fh  describe  a  semi- 
circle ;  in  which  (bhy  =  bh  X  bf.  Then  describe  the  arc  he 
from  h  as  a  centre,  which  will  give  be  =  x.  Then  make  be'  =  be 
and  ede'  an  equilateral  triangle  on  ec!,  which  will  give  the  three 
centres  required. 

For,  as  bf=  2  bh;  (bh)2  =  bhx  2bh  =  2(bhy 

and        (hh)2  =  (bhy  +  (bhy  =  3(bhy 

or,  (hk)  =  (»)vr=  ^^VT; 

then  be  =  bh  -\-  hh  =  — 1 „ —  \/  73]  =  x  ;  and 

r  —  a  —  x,  and  R  =  a  -f-  a;  as  required. 

45.  The  curve  in  Fig.  11  has  the  advantage,  in  application 
to  bridge  arches,  that  when  b  is  >  f  a,  it  affords  a  greater  in- 
terior capacity  for  the  flow  of  water  than  does  an  ellipse  on  the 
same  axes.     For  the  radius  of  curvature  at  a,  Fig.  17,  of  the 

b2 
ellipse  on  ab  and  bg  as  semi-axes,  is  rx  =  - ,  but 

(a  —  b    .    a  —  b      .-$■  \ 
— 2 r  — 2~  V3  ) 

Now  when  b  =  §  a, 

a         y 
but  r  =  (l  —  1+y  8>i  a  =  .545  a 


=  (l_l±VI)^, 


R 

3°.  Second  special  case.     The  ratio  -  to  be  a  minimum. 

r 

46.  From  Eq.  (2)  Art.  43,  —  =  -  +  h\~  2<T 

v   '  r  2r  (b  —  r) 

Differentiating  by  the  rule  for  fractions  (a  and  b,  constants) 
dividing  by  dr,  and  placing  the  result  =  0,  we  have 

7  1  R 

£  =__jr=  ar2  —  (a2  -f  P)  r  +  \b  (a2  +  b2)  =  0 ; 


20  STEREOTOMY. 

which  solved  with  respect  to  r  gives,  after  reducing,  and  neg- 
lecting that  value  of  r  which  makes  r  >  b, 

r  =  ygqrg  / vv+F-(«_-S)\  (6) 

Equating  this  with  (3)  and  reducing, 

R=  V^+V  /V?+g  +  («--ft)\  (6) 

The  direct  reduction  being  somewhat  tedious,  note  the  sym- 
metry of  Eqs.  (2)  and  (3),  where  (3)  is  obtained  by  substitut- 
ing a  for  b  and  R  for  r  in  (2)  ;  and  it  will  be  obvious  that  (6) 
is  obtained  from  (5)  by  a  like  substitution. 

47.  The  construction,  which  is  very  simple,  is  shown  in  PL 
II.,  Fig.  19.  Draw  the  chord  ae,  make  ef  =  a  —  b,  bisect  af 
by  the  perpendicular  gd,  meeting  eb  produced,  and  ab,  at  d  and 
c,  the  required  centres  for  the  half  curve  aoe.  For,  the  similar 
triangles  age,  aeb,  and  gde  give 

'^gi  +  b2—  (g  —  b)\ 

and  R=de  =  ~  X  eg  =  ^^  (vff  +  ^+ (« -- *),) 

Five-Centred  Ovals. 

4°.  To  construct  a  five-centred  semi-oval,  which  shall  con- 
form as  nearly  as  possible  to  a  semi-ellipse  on  the  same 
axes. 

48.  Five-centred  ovals  are  preferable  to  three  centred  ones, 

when  ~—  <  -j ;  and  are  generally  most  pleasing  when,  as  here 

required,  they  most  nearly  resemble  an  ellipse,  described  on 
the  same  axes.  In  PI.  II.,  Fig.  21,  let  fg  =  ^  ag.  It  is  a  prop- 
erty of  the  ellipse  that  its  radius  of  curvature,  at  the  extremity 
of  the  minor  axis,  is  a  third  proportional  to  the  semi-minor  and 
semi-major  axes.  Hence  make  fe  =  2ga  and  e  is  one  of  the 
five  centres.  Again,  the  radius  of  curvature  at  the  extremity 
of  the  major  axis,  is  a  third  proportional  to  the  semi-major  and 
semi-minor  axes,  hence,  make  ca  =  c'b  =  \fg<>  and  c  and  c'  will 
be  two  other  centres.  Now,  since  the  radius  of  curvature  of  an 
ellipse  is  changing  continually,  a  radius  may  be  found  which 
shall  be  a  mean  proportional  between  the  radii  already  found, 
and  such  a  radius  is  also  a  mean  proportional  between  the 
semi-axes,  for/e  :  ga: :  gf :  ae;  . • .  fe  X  ac  =  ga  X  gf  . 


■  ae  = 

de  = 

ae 
ab 
ae 
~be~ 

X 
X 

ag-. 

eg 

II           II 

2 +  6' 
a 

= 

z2  +  b* 
b 

STONE-CUTTING.  21 

that  is,  R  X  r  =      a  X  b,  and  calling 
the  intermediate  radius  rx,  we  have  by  hypothesis, 

r*  =  R  .  r.  Hence  r*  =  a  .  b  . 
that  is,  a  :  rx  :  :  rx  :  6. 
Hence  make  gj=gf,  describe  a  semicircle  on  aj ;  draw  #&, 
and  we  have  gk,  a  mean  proportional  between  ag,  =  a,  and 
#/,  ==  5,  and  hence  equal  to  the  radius  rx.  Now  make  if=gk  ; 
draw  the  arc  did' ;  make  ah  =  #&  ;  draw  the  arc  hd,  and  simi- 
larly draw  h'd' ;  then  ^  and  <#'are  the  remaining  centres;  for/g 
may  be  regarded  as  containing  all  the  lesser  radii  found  in 
moving  from  f  to  a  about  e  as  a  centre ;  i.  e.,  at  some  point 
of  the  motion  of  ef  about  e,  it  becomes  =  d  m.  Likewise  ab 
may  be  regarded  as  containing  all  the  radii  greater  than  ac, 
found  in  moving  ab  about  c  as  centre,  i.  e.,  at  some  point  of 
the  motion  of  ab  about  c,  ac  becomes  =  d  m,  hence  d  must  be 
at  the  intersection  of  arcs  di  and  dh,  drawn  with  e  and  c  re- 
spectively as  centres,  ah  and  fi  each  being  equal  to  rx.  A 
general  construction  of  the  five-centred  oval  is  given  in  my 
"  Drafting  Instruments  and  Operations,"  p.  92. 

5°.   To  construct  the  five-centred  oval,  by  a  method  ap- 
plicable to  an  oval  having-  any  number  of  centres. 

49.  See  PI.  II.,  Fig.  12,  where,  as  before,  put  ao  =  ar  and 
bo  =  b.  The  problem  is  indeterminate  when  the  extreme  radii 
R  and  r  are  both  chosen  arbitrarily,  for  calling  rx  =■  the  inter- 
mediate radius,  we  have  ec'  =  rx  —  r,  and  e"c'  =  H  —  rx ;  mak- 
ing but  two  equations  for  the  three  unknown  quantities  cc', 
c"c',  and  rL.  Note  that  as  cc'  -f-  c'c"  =  R  —  r,  the  centre  c' 
will  be  on  an  ellipse  whose  foci  are  c  and  c",  and  whose  major 
axis  is  R  —  r. 

In  order  to  render  the  problem  determinate,  put 

1°,  oc"  ==  3  oc,  i.  e. 
R  —  b  =  3  {a  —  r),  then,  2°,  ec"  =  |  oc",  and 

3°,    cd  ==  J  co. 

The  problem  can  now  be  solved  geometrically,  and  without 
assuming  any  of  the  required  radii,  thus.  Assume  ag ;  make 
og"  =  2>og  ;  bisect  og"  at  e';  join  g  and  e',  and  with  ^asa  centre, 
and  radius  ag,  describe  the  arc  aG.  Then  take  dg=-\  og ; 
draw  dg",  which  will  cut  ge'  in  a  point  g',  which  is  the  centre 
of  the  arc  GV ;  and  g"  is  the  centre  of  the  arc  VC.  In 
general,  however,  it  is  useless  to  draw  the  arcs  as  these  will 
not  be,  except  by  accident,  the  ones  required. 


22  STEREOTOMY. 

But  the  polygons,  ogg'g",  and  occ'c",  having  their  sides  made 
proportional,  will  always  be  similar,  hence  if  we  put, 

oc   =x;  og   =  p;  cc'-\-c'e"  =  z; 

oc"  =  y  ;  of  =  q  ;  gg'+g'g"  =  s, 
we  shall  have 

x       V     x       z  i  ii 

-  —  -:  _  =  -•  and  2-4-a  —  z=y-\-b, 

from  which  after  eliminating  z, 

p  +  q  —  s  *        p+q  —  s 

or,  when  as  was  assumed,  q=  3  p,  then  y  =  Sx,  and 


IV°.  —  Arches  —  Illustrations. 

Problem  IV. 
A  three-centred  arch  in  a  circular  wall. 

I.  The  Projections.  In  PL  III.,  Fig.  22,  a'a"d'd"  is  the 
plan  of  a  segment  of  the  wall,  4  feet  7  inches  thick,  of  a 
circular  room,  which  has  a  radius,  Cb  (  =  65")  of  25  feet  8 
inches.  This  wall  is  pierced  by  a  horizontal  cylinder,  whose 
axis  C"C  —  C,  intersects  the  vertical  line,  at  b,  that  of  the 
room,  at  b,  C,  and  whose  right  section  is  ABD  ;  forming  an 
arch,  whose  span,  AD,  is  17  feet  4  inches ;  rise,  CB,  is  6  feet 
2  inches  ;  interior  height,  GB,  is  17  feet  2  inches;  and  height 
to  top  of  keystone  is  20  feet  2  inches. 

The  face  line,  ABD,  is  found  by  (47)  the  joints  radiate  to 
the  centres  O,  c'  and  c",  and  are  adjusted  to  the  horizontal 
joints  of  eleven  equal  courses  of  stone,  each  22  inches  thick. 

From  these  numerical  data,  and  general  descriptions,  the 
drawings  can  be  made  (better  on  at  least  a  scale  of  ?l)  and, 
in  the  plan,  should  show  all  the  coursing  joints  (20)  of  one 
half  of  the  intrados,  as  e'"  is  shown  at  e4e5. 

II.  The  Directing  Instruments.  In  illustration  of  the 
derivation  of  these  from  the  projections  of  the  arch,  we  shall 
consider  some  one  stone.  Let  it  be  mnpqr.  As  usual,  the 
instruments  required  will  be  of  two  kinds :  patterns  to  deter- 


STONE-CUTTING.  Z6 

mine  the  forms  of  the  faces  of  the  stone  ;  and  bevels  to  deter- 
mine their  relative  positions  (7). 

1°.  The  pattern,  No.  3,  of  the  top,  pq,  will  be  the  figure 
P'P"Q'Q",  seen  in  its  real  size  in  the  plan. 

2°.  That  of  the  side  on  qr,  No.  4,  will  be  simply  a  rectangle, 
of  width  qr,  and  length  Q  Q". 

3°.  That  of  the  radial  bed  on  mr,  No.  7,  requires  the  con- 
struction of  the  true  form  of  the  elliptical  face-joint,  mr  —  M'Q'. 
As  this  construction  is  the  same  for  all  such  joints,  it  is  here 
given  for  them  all.  (See  Des.  Geometry,  Part  I,  Problem 
LXXXIX.,  2°.)  The  circle  of  radius  CK  =  bb"  =  bO,  is  the 
horizontal,  and  the  tangents  to  it,  as  KA,  are  the  vertical  pro- 
jection of  the  vertical  cylinder  of  revolution,  from  which  the 
face-joints,  mr,  np,  are  cut,  as  at  e'e,  e"g,  etc.,  by  planes  c'f 
c"h,  etc.,  perpendicular  to  \/,  and  oblique  to  the  axis  of  the 
cylinder.  These  planes  will  therefore  cut  the  cylinder  in 
ellipses  whose  semi-transverse  axes  are  c'f  c"h,  etc. ;  and 
whose  semi-conjugate  axes  are  each  equal  to  the  radius,  CK, 
of  the  cylinder. 

These  ellipses,  being  thus  known  by  their  axes,  each  may  be 
shown  first  by  revolving  it  into,  or  parallel  to  a  plane  of  pro- 
jection ;  or,  second,  all  may  be  shown  in  one  figure  constructed 
on  a  common  conjugate  axis. 

First  Method.  The  plane  of  np,  may  be  -revolved  to  the 
right  about  the  axis  ;  sb'"  —  np,  parallel  to  the  vertical  plane, 
when,  after  revolution,  6"'N'  will  appear  at  riN'" ;  kM.'  a>tjj", 
and  sP'  afcpp"  ;  all  perpendicular  to  np.  Then  p"j"W  will 
be  the  true  form  of  the  elliptic  arc,  N'P'  —  np,  forming  a  face 
joint.     Others  may  be  similarly  found. 

Second  Method.  Again,  in  Fig.  25,  lay  off  from  K',  on  K'H 
produced,  a  distance  =  KC,  from  Fig.  22,  giving  a  point  O. 
Then  on  a  perpendicular  to  K'O,  at  O,  lay  off,  from  O,  dis- 
tances to  the  left,  equal  to  the  semi-transverse  axes,  CK  (of  the 
circular  right  section  in  the  plane  CK)  c'f,  c"h,  etc.,  and  con- 
struct quarter  ellipses  on  K'O  and  these  several  semi-transverse 
axes.  Finally,  since  CD,  c'e',  c"e",  etc.,  are  the  perpendicular 
distances  of  the  inner  extremities,  D,  e',  e",  etc.,  of  the  joints 
from  the  conjugate  axes,  perpendicular  to  the  paper  at  C,  c',  c", 
etc.,  lay  off  these  distances  from  O  on  the  semi-transverse 
axes  in  Fig.  25,  giving  points  from  which  draw  ordinates,  par- 
allel to  K'O,  which  will  give  D,  e',  e",  e'",  etc.,  corresponding 


24  "  STEEEOTOMY. 

to  D,  e',  e"  e'",  etc.,  in  Fig.  22.  The  elements  of  the  intrados 
being  parallel  to  the  conjugate  axes,  at  C,  c',  etc.,  of  the  ellip- 
ses, Fig.  22 ;  aT)c,  de'f,  ge"i,  etc.,  Fig.  25,  are  bevels  giving 
the  true  forms  and  positions  of  the  face-joints  Dd,  e'e,  e"g, 
etc.,  Fig.  22,  relative  to  the  elements  of  the  intrados. 

Returning  to  the  joint  np  ;  pp" ,  jj",  and  nW",  being  revolved 
positions  of  lines  parallel  to  the  axis  of  the  arch,  if  we  lay  off, 
from  p",  j"  and  N",  on  these  lines  produced,  distances  equal 
to  N'N",  M'M"  and  P'P",  we  shall  have  the  true  form  of  the 
face-joint,  np  —  N"P",  and  of  the  pattern  No.  5,  of  the  radial 
bed  WWM'W!.  The  pattern  of  the  radial  bed  at  mr  may- 
be similarly  found. 

4°.  The  soffit  mn.  The  patterns  of  this,  and  of  all  the  other 
like  surfaces,  are  shown  on  the  development,  Fig.  28,  of  the 
entire  intrados,  ABD,  of  the  arch.  The  length  of  this  devel- 
opment, =  2AC,  Fig.  28,  =  2AmnB,  Fig.  22.  Projecting  over 
GI  from  M'M",  and  HJ  from  N'N",  and  proceeding  likewise 
for  the  other  stones,  we  get  GHIJ  as  the  pattern  No.  6,  of  the 
surface  mn  —  M'M"N'N",  for  example. 

5°.  The  patterns,  Nos.  8  and  9,  of  the  two  faces  of  the  stones 
will  be  found  at  once  by  developing  the  concave,  and  the  con- 
vex cylindrical  faces,  A'C'D',  and  A"C"D",  of  the  arch. 

6°.  Having  now  the  patterns  of  all  the  faces  of  the  stone  mn 
pq,  their  relative  position  may  be  determined  by  the  square  as 
shown  at  S ;  a  bevel  No.  10,  set  to  the  angle  npq,  and  an  arch 
square,  as  VY,  No.  11. 

7°.  Both  plane  beds  being  first  finished,  the  templet  vw,  No. 
13,  might  replace  No.  11. 

III.  Application.  This,  so  far  as  not  already  evident  from 
the  description  of  the  directing  instruments,  would  be  as  fol- 
lows. 

Taking  the  stone  just  considered,  the  radial  bed  on  np, 
would  first  be  wrought  by  No.  1,  the  straight  edge,  and  the 
pattern  No.  6.  Thence  the  top  could  be  wrought  by  No.  10, 
and  the  pattern,  No.  3  ;  and  the  soffit,  by  No.  11,  and  the  pat- 
tern, No.  6.  Also  No.  2,  held  in  planes  parallel  to  the  face  qr, 
will  give  elements  of  the  cylindrical  faces  of  the  stone  in  their 
true  relation  to  the  top  pq.  Or,  a  frame  TT',  No.  12,  whose 
parallel  bars  TT  and  T'T'  are  curved  to  the  radius  of  the  face, 
and  held  in  planes  parallel  to  the  top,  pq,  would  give  circular 
lines  of  the  face,  between  which  No.  1,  held  perpendicularly  to 


STONE-CUTTING.  25 

TT,  would  test  the  proper  cutting  away  of  the  intermediate 
rough  stone. 

Examples.  —  1°.  Substitute  for  the  circular  wall,  shown  in  the  plan,  a  straight 
wall,  with  a  sloping  face,  as  shown  in  Fig.  24,  and  make  the  isometrical  drawing 
of  any  one  of  the  stones  ;  as  indicated  in  Fig.  26  when  the  batter  is  on  the  front. 

Ex.  2°.  Make  the  isometric  drawing,  Fig.  23,  of  a  stone  from  the  cylindrical 
wall. 

Ex.  3°.  Construct  any  of  the  arches  already  described,  on  Pis.  II.  and  III.  on  a 
large  scale  ;  with  an  isometrical  drawing  of  any  one  of  the  stones  ;  also  a  develop- 
ment of  all  the  faces  of  one  stone  into  the  plane  of  the  paper. 

Ex.  4°.  Do.  for  PI.  III.,  Fig.  27. 

Ex.  5°.  Do.  for  PI.  III.,  Fig.  29,  supplying  a  plan,  and  batter  at  one  end. 

Ex.  6°.  In  Prob.  IV.  let  the  face  of  the  arch  be  in  a  vertical  plane,  but  oblique  to 
the  axis  of  the  arch. 

Ex.  7°.  Let  the  arch,  either  segmental  or  full  centred,  be  in  a  recess,  with  diver- 
gent sides,  as  in  the  plate  band,  Prob.  III. ;  and  therefore  conical  above  the  spring- 
ing lines. 

Pkoblem  V. 

A  semi-cylindrical  arch,  connecting  a  larger  similar  gallery, 
perpendicular  to  it,  on  the  same  springing  plane ;  with  an  in- 
closure  which  terminates  the  arch  by  a  sloping  skew-face. 

I.   The  projections.  PI.  IV.,  Figs.  30-32. 
1°.  Let  the  following  be  the  given  dimensions. 
Skew  (24)  of  the  oblique  wall  PQC4,  =  18°. 
Batter    "         "         »  "        "        =  T^. 

Radius,  O'A',  of  intrados  of  arch  =  3'. 

"       O'a'     "  extrados  "    "  =  4'  :  6". 

"       O'K,  "  intrados  of  gallery       =  6'  :  6". 
Least  thickness  of  wall  =  2!  :  3". 

Greatest  "  "     "  =  5'  :  4". 

Let  the  springing  plane  be  taken  as  the  plane  H>  and  let  the 
given  thicknesses  of  the  wall  be  in  it ;  and  let  the  plane  V  he 
perpendicular  to  the  axis,  0"0  —  O',  of  the  required  arch. 
Let  there  be  five  voussoirs,  dividing  the  section  A'E'B',  of  the 
intrados  equally,  and  let  them  be  completed  by  horizontal  and 
vertical  planes,  as  CD'  and  G'C,  through  the  outer  extrem- 
ities, D'  and  G',  of  the  radial  beds. 

Then,  with  a  scale  of  not  less  than  £%,  =  2'  to  1",  in  order  to 
be  more  easily  accurate,  the  given  dimensions  can  be  drawn,  as 
shown,  where  JR  is  the  horizontal  trace  of  the  vertical  side  of 
the  wall. 

JR  —  KR'  is  one  springing  line  of  the  gallery,  all  of  whose 


26  STEEEOTOMY. 

elements  are  therefore  parallel  to  JR,  in  front  of  the  vertical 
plane  JR,  and  above  the  plane  H5  of  the  springing  lines,  JR  — 
KR' ;  and  AA"  —  A',  and  BB"  —  B',  of  the  arch. 

2°.  Declivity  of  the  plane  face  of  the  arch.  —  PQ,  at  18°  with 
JR,  is  the  horizontal  trace  of  the  plane  of  this  face.  Its  bat- 
ter, ^\  (3  to  10),  is  perpendicular  to  PQ.  Hence,  assume  hh, 
perpendicular  to  PQ,  revolve  it  about  a  vertical  axis  at  b,  to 
5L",  parallel  to  V5  make  b'p'  =  10,  from  any  convenient  scale 
of  equal  parts,  and  p'L'  =  3,  from  the  same  scale  ;  and  \Jb'  will 
be  the  revolved  position  of  the  line  of  declivity,  5L,  showing 
the  real  slope  of  the  plane  arch  face. 

From  this  batter  we  find  next,  for  convenience  in  projecting 
points,  the  slope,  taken  in  vertical  planes  parallel  to  the  axis, 
0"0 —  O'.  Thus  LI,  parallel  to  PQ,  is  the  horizontal  projec- 
tion of  a  horizontal  line  in  the  plane  face  of  the  arch.  Note  I, 
its  intersection  with  the  vertical  plane  bb",  revolve  I  to  I",  pro- 
ject it  to  F,  onp'U,  and  lib'  is  the  vertical  projection  of  lb, 
and  is  the  declivity  of  the  plane  face  of  the  arch,  in  the  vertical 
plane  bb". 

3°.  Horizontal  projection  of  the  plane  face.  —  Draw  horizon- 
tals, through  all  the  points  of  the  face,  as  G'Cj  through  C  and  D' ; 
produce  them  to  meet  Vb',  as  at  C2 ;  then,  for  instance,  project 
C2  at  C3,  and  revolve  it  to  C4 ;  then  C4  C,  parallel  to  PQ,  will 
be  the  horizontal  projection  of  C'C^  and  will  intersect  the  pro- 
jecting lines,  from  C  and  D',  at  C  and  D,  the  horizontal  pro- 
jections of  C  and  D'.     Find  other  points  similarly,  or  — 

Otherwise  :  project  k'  at  k"  and  revolve  it  to  k,  when  kC 
will  coincide  with  C4C,  whence,  as  before,  etc. 

Again :  as  C1C2  is  the  true  distance  of  C  and  D'  in  front  of 
the  vertical  plane  on  PQ,  and  in  the  direction  of  the  axis  0"0 
■ —  O',  make  iD  —jC  ==  C1C2,  and  we  have  C  and  D  as  before. 

The  horizontal  projections  of  the  radial  joints  all  meet  at  O. 

The  semi-elliptic  face  line,  AEB,  has  AB  and  20S  for  a  pair 
of  conjugate  diameters  ;  hence  it  is  tangent  to  A  A"  and  BB"  at 
A  and  B,  and  at  S  has  a  tangent  parallel  to  AB. 

50.  The  last  of  the  three  constructions  of  CD,  etc.,  just  given, 
namely,  by  the  method  of  compass  transference  of  knotvn  dis- 
tances is  advantageous,  in  avoiding  numerous  lines  of  con- 
struction, as  all  from  C2  to  C ;  yet  for  the  same  reason,  disad- 
vantageous, in  not  preserving  upon  the  paper  such  traces  of  the 
construction  as  would  enable  any  one  to  recall  it  from  the 
drawing  alone. 


STONE-CUTTING.  27 

4°.  Horizontal  projection  of  the  cylindrical  face.  —  JKJ'  is  a 
profile  plane,  which  contains  a  semicircular  right  section  of  the 
cylindrical  gallery.  Revolving  this  plane  about  a  vertical  axis 
at  J,  the  centre  of  that  section  will  appear  as  at  Ox  by  making 
KOx  equal  to  the  given  internal  radius  of  the  gallery ;  and  Kc2, 
with  Ox  as  a  centre,  will  be  the  revolved  position  of  the  section. 
Thence  the  horizontal  projection  of  any  points  of  the  cylindri- 
cal face  of  the  arch  can  be  found  as  before. 

Thus,  produce  D'C  to  c1?  and  then  either  make  cC"  =  cxc^ 
the  true  distance  of  C"C  in  front  of  the  vertical  plane  on  JR ; 
or,  by  showing  the  counter  revolution,  etc.,  project  c2  on  JR,  at 
c3,  not  shown,  counter  revolve  cs  to  c4,  not  shown,  on  KJ  pro- 
duced, whence  project  it  by  a  line  parallel  to  JR,  till  it  meets 
CC",  giving  C".  In  like  manner  all  points  of  the  cylindrical 
face  may  be  found. 

The  radial  joints  D"E",  etc.,  of  this  face  are  arcs  of  ellipses, 
being  sections  of  the  cylindrical  intrados  of  the  gallery  by  the 
planes  OO'D',  etc.,  which  cut  it  obliquely. 

Opposite  joints,  symmetrical  with  00",  as  D"E"  and  dxex. 
form  parts  of  one  ellipse,  D"0"c?!,  in  horizontal  projection, 
since  dxex  —  d\e\  is  exactly  over  that  part  of  the  ellipse  D"0" 
—  D'O'  whose  vertical  projection  is  on  D'O'  produced. 

All  the  lines  of  the  cylindrical  face  are  invisible,  and  hence 
dotted,  except  such  top  and  lateral  edges,  as  CD"  and  C"G". 

II.  The  directing  Instruments.  —  We  may  either  show  to- 
gether the  patterns  of  like  faces  of  all  the  stones,  or  the  pat- 
terns of  all  the  faces  of  one  stone. 

Adopting  the  latter  method  as  clearer,  while  illustrating  all 
the  operations  required  by  the  former,  let  the  stone  C'D'F'  be 
chosen  for  detailed  representation. 

Development  of  the  stone  C'D'F'.  —  A  right  section  of  this 
stone  is  the  polygon  CD'E'F'G',  which  will  develop  in  a  straight 
line  as  AiBx,  Fig.  31. 

Then,  supposing  the  top  face  to  be  the  plane  of  develop- 
ment, and  that  the  faces  to  the  right  of  D',  around  to  G',  are 
developed  to  the  right  of  the  edge  D"D  —  D',  while  the  face 
C'G'  revolves  to  the  left  about  the  edge  C"C  —  C  into  the 
plane  CD'  ;  we  shall  make,  in  Fig.  31,  Axc  ;  cd  ;  de  ;  ef;  and 
fB1  equal  respectively  to  G'C  ;  CD' ;  D'E'  ;  E'F,  and  F'G'  in 
Fig.  30.     At  these  points  draw  lines  perpendicular  to  A^, 


28  STEREOTOMY. 

for  the  indefinite  developments  of  those  edges  of  the  voussoir 
which  are  parallel  to  the  axis,  0"0  —  O'. 

To  develope  the  edges  of  the  plane  end  of  the  stone.  —  Lay  off 
the  perpendicular  distances  of  their  extremities  from  some 
plane  of  right  section,  as  JR.  Thus  Axgx  =  cG  ;  cCx  =  cC, 
etc.,  the  second  term  of  each  equality  being  taken  from  the 
plan  in  Fig.  30.  The  curved  edge,  EF — E'F',  is  developed 
by  taking  one  or  more  intermediate  points,  as  HH7,  and  mak- 
ing eA,  and  7iHj,  Fig.  31,  respectively  equal  to  E'H7,  and  H^, 
in  Fig.  30.  Also,  for  additional  accuracy,  make  ^02  =  D'0', 
and  OA  =  0"0,  then  Ox,  Fig.  31,  will  be  a  point  of  D^  pro- 
duced. Likewise,  make  B^  and  o2ox,  Fig.  31,  ==  G'O',  and 
0"0,  Fig.  30,  and  ox  will  be  a  point  on  G^  produced. 

To  develope  the  edges  of  the  cylindrical  face  of  the  same  vous- 
soir. —  As  before  in  Figs.  31  and  30,  respectively,  Axg2  =  eG"  ; 
cC2  =  cC"  ;  Nrc  =  Nw  ;  R2h  =  W\,  etc.  Also,  as  C"G"  — 
C'G'  is  an  arc  of  the  right  section  of  the  gallery,  C2g2,  Fig.  31, 
is  drawn  with  a  radius  ==  OxK,  Fig.  30,  and  laid  off  on  the 
perpendicular  at  the  middle  of  the  chord  C2g2.  The  elliptic  arc 
D"E"0"  is  developed  in  its  real  size  at  D2E202 ;  and  as  every 
edge  of  the  cylindrical  face  is  curved,  except  C2D2,  they  will  be 
curved  in  the  development,  and  must  be  there  found  by  inter- 
mediate points,  as,  at  Mm,  =  Mm,  Fig.  30. 

To  develope  the  plane  face  of  the  voussoir.  —  Revolve  it  about 
CD  —  CD'  into  the  plane  of  the  top.  The  true  lengths  of  G#, 
F/,  and  Ee,  Fig.  30,  found  at  k'r,  k's,  and  k't,  on  the  line  of 
declivity  L'5',  will  then  be  laid  off  on  ^G,  /XF,  and  ^E,  per- 
pendicular to  C^Dj,  Fig.  31 ;  where  g,  fx  and  ex  are  transferred, 
as  shown,  from  CD,  Fig.  30,  to  its  equal,  CXT>X,  Fig.  31.  The 
true  size  of  the  radius  EO — E'O'  is  found  by  revolving  it 
about  OO"  into  the  plane  H,  at  e'"0.  That  of  FO  —  F'O'  is, 
likewise, /"'0  ;  then  arcs  from  E  and  F,  Fig.  31,  as  centres, 
with  these  radii,  respectively,  will  give  03,  the  development 
of  the  centre  of  the  ellipse  of  which  the  face  line,  EF,  is  a  part, 
and  the  point  to  which  GF  and  DXE  radiate. 

To  develope  the  cylindrical  face  of  the  voussoir.  —  Here  Fig. 
32,  F/,  for  example,  is  the  true  length,  taken  from  c2/2,  Fig.  30, 
of  the  circular  arc  whose  vertical  projection  is  F'f.  Other 
points,  being  similarly  found  and  joined,  give  the  pattern,  which 
must  be  flexible,  of  the  cylindrical  face.  When  flat,  all  its 
edges  are  curved  except  CD  and  GG.     The  developed  faces  of 


STONE   CUTTING. 


the  voussoir  are  the  patterns  of  those  faces, 
voussoirs  would  be  similarly  found. 


29 

Those  of  the  other 


III.  Application. — Select  a  stone  whose  right  section  will 
circumscribe  C'D'F'G',  and  whose  length  shall  not  be  less  than 
Ee'',  and  work  the  top  by  No.  1,  the  straight  edge  (7)  and  No. 
3.  Make  the  left  side  square  with  the  top  by  No.  2,  and  finish 
it  by  its  pattern.  No.  4,  with  the  pattern  of  the  bed  at  D'E', 
will  complete  that  face.  The  arch  square,  No.  5,  with  the  pat- 
tern of  the  intrados,  will  give  the  surface  EE"FF"  ;  and  in  like 
manner  the  whole  convex  surface  can  be  wrought.  All  the 
edges  of  both  ends  of  the  stone  being  thus  given,  their  patterns, 
with  the  straight  edge,  will  serve  to  complete  them. 

Otherwise :  the  plane  and  cylindrical  ends  can  be  wrought 
next  after  the  top,  by  bevels  Nos.  6  and  7,  respectively. 

This  problem  includes  the  following  simpler  cases,  added  as 
examples. 

Examples.  —  1°.  Let  both  ends  be  plane;  one,  vertical  on  PQ  ;  the  other  ver- 
tical on  JR. 

Ex.  2°.  Let  the  cylindrical  end  be  replaced  by  a  plane  one,  having  JR.  for  its 
horizontal  trace,  and  a  batter  of  ^. 

Ex.  3°.  Construct  a  rampant  (ascending)  arch  covering  a  straight  flight  of 
steps. 

Grroined,  and  Cloistered  Arches. 

51.  Both  of  these  kinds  of  arches  are  compound,  being 
formed  by  the  intersection  of  two  single  arches,  each  of  which 


Fig.  3. 


30  STEREOTOMY. 

is  usually  cylindrical.  They  are  distinguished  from  each  other 
as  follows  :  — 

In  the  groined  arch,  Fig.  3,  that  part  of  each  cylinder  is  real 
which  is  exterior  to  the  other.  Thus  EF,  LM,  eF,  and  KH  are 
real  portions  of  elements. 

In  the  cloistered  arch,  that  part  of  each  cylinder  is  real  which 
is  within  the  other.  Thus  in  Fig.  3,  HF  and  FM  would  be  the 
real  portions  of  the  elements. 

The  groined  arch  therefore  naturally  covers  the  quadrangular 
space  at  which  two  arched  open  passages,  ABC  and  abc,  inter- 
sect ;  while  the  cloistered  arch  forms  the  doubly  arched  cover, 
or  quadrangular  dome,  of  a  quadrangular  enclosed  room,  or 
cell. 

Theorem  I. 

Saving  two  cylinders  of  revolution,  whose  axes  intersect,  the  pro- 
jection of  their  intersection,  upon  the  plane  of  their  axes,  is  a 
hyperbola. 

See  PI.  IV.,  Fig.  30,  where  020  —  O'  is  the  axis  of  the  cyl- 
inder A'S'B',  and  a  parallel  to  JR  through  02,  where  0"02  = 
OxK,  is  the  axis  of  the  cylinder  all  of  whose  elements  are  par- 
allel to  the  one  JR. 

A"S"B"  is  the  horizontal  projection  of  the  intersection  of 
these  cylinders,  that  is,  upon  the  plane  of  their  axes,  and  it 
will  be  shown  to  be  a  hyperbola  by  proving  that  (02h"y — 
(R"h"y  =  (02S")2 ;  where  H"  is  any  point  on  A"S"B"  and 
H"h"  a  perpendicular  (ordinate)  to  the  axis  020. 

Suppose  (02h»y  —  (H" A")2  =  (02S")2.  (1) 

Now  (H"A")2  =  (R'h"')*  =  S'h'"  X  sh">  ;  where.  S'«  =  A'B'. 

And  S'h'"  X  sh'"  =  S'h'"  X  (S'O'  +  O'A"') 
=  S'h'"  X  (S2s2  +  H2A2). 

But  (S2s2y= (OA)2  —  (o^)2. 

And  (HsAs)8  =  (01Sa)2  —  (Oxh2y. 

Whence,  since  Oi«2  =  02S",  and  01h2  =  02h" 

(s2s2y  —  (H2h2y=(02h»y—  (02s»y. 

But  also  (S2s2)2  —  (H2A2)2  =  (O'H')2  —  (O'A'")2  =  (K'h">y 
=  (E"h"y. 

Hence  (02A")2  —  (02S")2  =  (H"A")S. 

Or,  (02h«y  —  (H"h"y  =  (o.s"y. 

Thus  equation  (1)  is  proved ;  and,  calling  02h"  =  x ;  and 


STONE-CUTTING.  31 

H"h"  =  y  ;  and  02S"  =  a  ;  we  have  x%  —  y2  =  a2 ;  which  is 
the  equation  of  an  equilateral  hyperbola,  that  is,  one  whose 
axes  are  equal.     Thus  the  theorem  is  proved. 

52.  An  important  consequence  of  the  last  theorem  is,  that 
when  the  cylinders  become  equal,  the  vertices  of  the  curve,  of 
which  S"  is  one,  unite  at  02.  Now  when  the  two  vertices  of  a 
hyperbola  coincide,  the  curve  reduces  to  the  special  case  of  two 
intersecting  straight  lines. 

But  the  actual  intersections  of  two  cylinders  (whose  axes  are 
not  parallel)  are  curves.  Hence  if  any  projection  of  these 
curves  is  straight,  they  are  plane  curves,  and  hence  ellipses. 

53.  A  little  consideration  of  the  properties  of  hyperbolas 
will  sufficiently  show,  what  there  is  not  room  here  to  strictly 
prove  :  1st,  that  a  change  in  the  angle  between  the  axes  of  the 
cylinders  would  only  cause  the  curve  A"S"B"  to  become  a  hy- 
perbola referred  to  the  new  axes  as  conjugate  diameters ; 
and  2d,  that  the  substitution  of  elliptic  for  circular  cylinders 
would  only  yield  a  general,  in  place  of  the  equilateral  form  of 
the  hyperbola.  The  conclusion  of  (52)  is  therefore  true  of  all 
cylinders  whose  diameters,  measured  in  a  plane  perpendicular 
to  that  of  their  axes,  are  equal. 

54.  Cylinders,  situated  as  just  described,  will  have  a  pair 
of  common  tangent  planes  parallel  to  that  of  their  axes.  This 
fact,  added  to  the  last  two  articles,  affords  several  statements  of 
what  is  really  the  same  proposition,  each  statement  being  ap- 
propriate to  certain  given  conditions.     Thus  — 

1°.  If  two  ellipses  intersect  in  a  common  semi-axis,  of  the 
same  kind  for  each,  their  other  axes  being  in  the  same  plane  P 
(thus  if  two  ellipses  whose  transverse  axes  bisect  each  other 
in  H  have  a  common  vertical  semi-conjugate  axis),  lines  join- 
ing points,  which  are  on  the  two  ellipses,  and  which  are  at 
equal  distances  from  P,  will  be  parallel,  and  will  therefore  form 
a  pair  of  cylinders,  of  which  these  ellipses  will  be  the  intersec- 
tions. 

2°.  If  two  cylinders  intersect  in  one  plane  curve,  as  an 
ellipse,  there  will  be  a  second  branch  of  the  intersection,  which 
will  also  be  plane. 

3°.  If  two  cylinders  whose  axes  are  in  the  same  plane  P, 
also  have  two  common  tangent  planes,  parallel  to  P,  they  will 
intersect  in  two  plane  curves,  which  will  cross  each  other  at 


32  STEREOTOMY. 

the  intersections  of  the  elements  of  contact  of  the  tangent 
planes. 

Problem  VI. 
The  oblique  groined  arch. 

55.  Design.  —  Suppose  that,  in  the  collecting  system  of  cer- 
tain water-works,  supplied  by  several  ponds,  two  conduits,  each 
covered  by  semi-cylindrical  arches  of  nine  feet  span,  unite  at 
an  angle  of  67°  :  30'  and  discharge  into  one  of  twelve  feet  span, 
covered  by  a  semi-elliptic  arch,  having  the  same  rise  and  spring- 
ing plane  as  the  former  ones. 

56.  We  may  note  in  passing  that,  supposing  the  water  to  be 
four  feet  deep  in  each  of  the  nine-foot  conduits,  the  sum  of 
their  water  sections  is  72  square  feet.  Then,  in  the  large  con- 
duit, if  the  water  be  but  four  feet  deep,  this  conduit  should  be 
18  feet  wide.  Or,  if  but  12  feet  wide,  as  in  the  problem,  its 
floor  should  be  sunk  so  that  the  water  in  it  should  be  six  feet 
deep ;  or  else  its  declivity  should  be  increased  to  give  such  a 
velocity  to  the  water  in  it  that  its  section  of  48  square  feet 
would  transmit  as  much  water  per  minute  as  passes  the  sec- 
tions of  36  square  feet  each,  of  the  two  nine-foot  conduits. 

I.  The  Projections.  —  Three  planes  of  projection  are  used: 
the  horizontal  plane,  H?  containing  the  springing  lines,  Hf  and 
YN,  YM  and  Y"M",  of  the  two  smaller  conduits,  and  HH'X  and 
H"H2,  of  the  larger  one ;  a  vertical  plane  V?  whose  ground  line 
is  P'Q',  and  which  is  perpendicular  to  the  axis  OX  —  X'  of  one 
of  the  smaller  conduits  ;  and  a  vertical  plane  Vn  whose  ground 
line  is  H1H2,  and  which  is  perpendicular  to  the  axis  Om  —  0{ 
of  the  larger  conduit. 

The  lines  in  the  horizontal  plane  can  be  first  laid  down  from 
the  given  dimensions  and  axes,  OX  and  Om,  of  the  arches ; 
giving  the  four  springing  points  H,  Y,  Y",  H",  of  the  groin. 
Then  OY  and  OY"  are  the  horizontal  projections  of  the  quar- 
ter ellipses  (52),  in  which  the  arch  whose  axis  is  OR  inter- 
sects the  right  hand  half  of  the  one  whose  axis  is  OX.  Like- 
wise OH  and  OH"  are  the  projections  of  the  intersections  of 
the  arch  whose  axis  is  Om  with  the  left  hand  half  of  the  one 
whose  axis  is  OX. 

Next  make  the  elevation  on  P'Q',  making  the  spandril  stones 
40"  thick,  the  thickness  O'l'  at  the  crown  3  ft.,  and  the  radius, 


STONE-CUTTING.  33 

q'V,  of  the  extrados,  9  ft.  The  radial  joints  from  X'  divide  the 
intrados  into  five  equal  parts. 

The  elevation  on  H^H^  is  now  made  as  follows :  — 

1°.  The  face  line  Hi  O"  W2.  —  Any  point,  as  a',  is  the  ver- 
tical projection  of  the  element  A" Am,  or  of  any  point  of  it. 
Hence  a'  is  horizontally  projected  upon  the  groin  curves  at  A 
and  A".  The  projections  of  these  points  on  Vi  will  then  be 
at  a[  and  a",  at  heights  above  H^Hg  equal  to  that  of  a'  above 
PQ. 

2°.  Other  points  of  the  elevation  on  Vi*  —  As  the  extrados 
is  seldom  a  finished  surface,  we  need  not  construct  the  groin 
curves  of  the  extrados,  but  may  proceed  as  follows,  to  find  ex- 
tremities of  radial  joint  lines  ;  these  joints  being  normal  to  the 
intrados  in  both  elevations. 

To  find  f\  for  example.  Draw  the  tangent  g'V  perpendic- 
ular to  X'y,  project  V  at  I  in  the  vertical  plane  OH  of  the  groin, 
when  Gl  will  be  the  horizontal  projection  of  g'V ',  considered  as 
the  tangent  to  the  groin  HO  at  Gg'.  Then  projecting  G  at 
g[  and  I  at  l[,  gives  g[l{  as  the  new  vertical  projection  of  this 
tangent,  and  mg{,  perpendicular  to  it,  as  the  like  projection  of 
the  joint  in  the  face  of  the  main  arch,  corresponding  to  X.'g'  on 
that  of  the  arch  H'O'Y'.  Finally  f[  is  the  intersection  of  the 
joint  mg[  with  e\f{  parallel  to  the  ground  line  HiH^,  and  at  a 
height  from  it  eiP"  =  e'P'. 

In  like  manner,  other  points  may  be  found,  as  may  be  seen 
at/i,  extremity  of  an  auxiliary  joint  X'/' —  mlj'i. 

3°.  The  curve  b"I'"ei,  right  section  of  the  extrados  of  the 
elliptic  arch  is  not,  as  might  be  supposed,  an  ellipse,  derived 
from  b'j'  as  0"Hi  was  from  OH  ;  since  the  two  conditions  of 
normal  joints,  and  equal  heights  of  like  joints  on  both  eleva- 
tions prevent  this. 

That  is,  i'i,  for  example,  determined  from  i'  by  projecting 
i'  upon  DO  as  the  straight  horizontal  projection  of  the  outer 
groin,  and  thence  to  i[,  as  g[  was  found  from  g'  would  not  co- 
incide with  i'x  of  the  figure,  found  on  the  given  normal  mgx  and 
at  the  same  height  as  i' ;  since  each  determination  is  complete 
in  itself  and  hence  independent  of  the  other. 

Points,  as  B,  C,etc,  of  the  horizontal  projection  of  the  outer 
groin,  are  at  the  intersection  of  projecting  lines  from  b'  and  b{ ; 
c'  and  c[,  etc.  ;  and  are  not  in  a  straight  line  with  O ;  thus 
showing  that  the  outer  groin  is  not,  in  space,  a  plane  curve. 

3 


34  STEREOTOMY. 

4°.  The  Projections  of  a  Stone.  —  The  stones  of  a  groined 
arch  in  jointed  masonry  are  partly  in  each  arch.  Hence,  tak- 
ing the  most  irregular  stone  as  an  example,  it  is  that  whose  sec- 
tion in  the  front  elevation  is  a'b'c'd'e'f'g'.  The  side  elevation 
of  the  end  in  the  elliptic  arch  is  a[b[d'1e/ig[i  and  its  plan  is  lim- 
ited by  the  figure  AadDd^. 

II.  The  Directing  Instruments.  —  Passing  by  Nos.  1  and  2, 
the  straight-edge  and  square,  which  are  of  constant  use  (7), 
there  are  the  following  :  — 

No.  3,  =  the  pattern  a'b'd1 .  . . .  g'  of  the  end  in  the  vertical 
plane  at  ad. 

No.  4,  =  the  pattern,  dcCcid^  of  the  plane  portion  of  the  top. 

No.  5  =  the  pattern  Ffdd^  of  the  plane  portion  of  the 
under  side. 

Fig.  34  shows,  together,  the  patterns,  Nos.  6,  7,  8,  and  9,  of 
the  principal  lateral  faces  within  the  circular  arch.  There, 
ab  =  a'b' ;  bc=  the  arc  b'c' ;  ag  =  the  arc  a'g'  ;  and  fg  =f'g'. 

Also  Cc,  Bb,  Aa F/  =  Cc,  Bb etc.,  in   the   plan, 

Fig.  33.  Thus  No.  6  is  the  pattern  of  the  plane  radial  bed 
ABab  —  a'b' ;  No.  7,  of  the  cylindrical  extrados  BCbc  —  b'c' ; 
No.  8,  of  the  cylindrical  intrados,  AGag  —  a'g' ;  and  No.  9,  of 
the  plane  radial  bed,  FG/# — /'#'• 

Detailed  description  of  the  analogous  patterns,  Nos.  10,  11, 
12,  and  13,  Fig.  35,  of  the  corresponding  lateral  faces  of  that 
part  of  the  stone  lying  in  the  elliptic  arch,  is  unnecessary,  as 
the  construction  of  Fig.  35  is  entirely  similar  to  that  of  Fig.  34. 

Thus,  atp^  Fig.  35  =  a\p{  from  the  elliptical  elevation; 
and  Aai  Ppx,  etc.,  Fig.  35,  =  Aa1  Ppl5  etc.,  from  the  plan  ; 
hence  AGa^i  is  the  pattern  of  the  soffit  AGa^  —  a\g\. 

As  a  further  aid  to  accuracy,  there  should  be  one  or  more 
right  section  bevels,  as  Nos.  14  and  15,  to  test  the  relative  po- 
sition of  the  lateral  faces.  No.  16  is  the  pattern  of  the  end  in 
the  plane  a^. 

III.  Application.  —  Choosing  a  block  of  the  thickness  hk, 
and  in  the  plan  of  which  Aaddxax  can  be  inscribed,  first  work 
the  end  in  the  plane  ad,  and  complete  it  by  the  pattern  No.  3. 
From  this,  by  means  of  No.  2,  the  square,  determine  the  direc- 
tion of  all  the  lateral  faces,  including  the  upper  and  under 
ones  ;  and  the  back,  whose  horizontal  projection  is  Dd  and 
Ddx ;  and  finish  them  by  their  patterns,  Nos.  4-9,  and  the  arch- 
square,  No.  14. 


STONE-CUTTING.  35 

This  done,  the  three  edges,  Cidx,  fxdv  and  d1  —  e'xd[  of  the  end 
in  the  plane  axdx  will  be  known,  whence  this  end  can  be  wrought 
square  with  the  top  and  bottom  plane  surfaces,  and  completed 
by  its  pattern,  No.  16.  Thence  the  remaining  lateral  surfaces 
within  the  elliptic  arch  can  be  directed  by  the  square,  and  com- 
pleted by  their  patterns,  Nos.  10-13,  and  No.  15. 

Examples. —  1°.  Let  the  cylinders  (having  a  common  springing  plane,  and 
equal  heights  in  every  case)  be  equal. 

Ex.  2°.  Let  them  be  at  right  angles. 

Ex.  3°.  Construct  the  patterns  for  the  key-stone  (which  will  extend  in  one  piece 
from  0,  on  all  the  cylinders). 

Ex.  4°.  The  arches  being  at  right  angles,  let  the  circular  cylinder  be  real  be- 
tween OH  and  OH",  and  the  elliptic  one,  between  OH  and  OY,  forming  a  clois- 
tered arch. 

Ex.  5°.  In  Ex.  4°,  construct  the  patterns  for  the  groin  stone  corresponding  to 
the  one  described  in  the  groined  arch  (Aa  will  be  real  from  A  forward,  and 
A«i  will  be  real  from  A  to  the  right). 

Ex.  6°.  In  Exs.  2°  and  4°,  let  the  extrados  be  finished  as  in  PI.  IV.,  Fig.  30. 


Problem  VII. 

The  groined  and  cloistered,  or  elbow  arch. 

57.  This  problem  is  designed  to  illustrate  very  compactly, 
that  is,  without  repetition  of  similar  parts,  first,  the  difference 
between  a  groined  and  a  cloistered  arch  ;  and,  second,  the  mode 
of  proceeding  when  it  is  determined  that  the  intersection  of  the 
two  extrados,  that  is,  the  outer,  as  well  as  the  inner  groin  curve, 
shall  be  a  vertical  ellipse. 

I.  The  Projections.  —  A  gallery,  PL  VI.,  Fig.  47,  whose  width 
A'B'  is  8  ft.,  intersects  one  of  10  ft.  in  width,  each  of  them  end- 
ing at  its  intersection,  ADB,  with  the  other.  These  galleries 
are  covered  by  arches  of  equal  rise,  CD'  =  CD",  and  whose 
axes,  CD  and  C"D,  are  in  the  same  plane.  They  therefore  in- 
tersect in  a  vertical  ellipse  (52),  whose  horizontal  projection 
is  ADB.     The  radius,  E'Ci,  of  the  given  extrados  is  5'  :  3". 

The  portion  A'C'DCA"  thus  forms  one  quarter  of  a  right- 
angled  groined  arch  ;  while  B'C'DC'B"  forms  one  quarter  of 
a  right-angled  cloistered  arch.  The  construction  will  mostly 
be  obvious  on  inspection,  by  comparison  with  PL  IV.,  Fig.  33. 

A  stone  of  the  cloister.  —  MPRr — M'P'  is  its  circular  intrados ; 
OPRQ— O'P'isa  radial  joint;  NOQ?  —  N'O' is  its  circular 
extrados;    MN^r — M'N'  is  another  radial  joint,  and  MO  — 


36  STEBEOTOMY. 

M'N'O'P'  is  its  vertical  plane  end.  The  like  surfaces  in  the 
elliptic  portion  are  obvious  by  reading  the  drawing. 

The  proposed  elliptical  outer  groin  may  coincide  with  the 
inner  one  AB,  in  plan.  That  is,  both  may  be  in  the  same 
vertical  plane.  Then,  taking  the  joint  at  a'c'  for  illustration, 
d  will  be  horizontally  projected  at  n  instead  of  at  c  ;  and  n  will 
thence  be  projected  at  n" ,  at  the  same  height  as  c' ;  likewise 
F,  at/,  on  BA  produced,  and  at/"  ;  and  so  on  for  other  points. 
Then  Wn"fn  will  be  an  ellipse,  it  being  the  right  section  of 
the  cylinder  whose  oblique  section  is  the  vertical"  ellipse  T)nf. 
Producing  the  normal  joint  a"c"  to  the  new  extrados  Wn"f", 
we  find  its  new  outer  extremity  o"  ;  and  the  other  vertical  pro- 
jection, o'o'",  at  the  same  height  as  o",  of  the  element  of  the 
new  extrados,  containing  o".  But  o'"o'  must  be  limited  by  the 
plane,  CV,  of  the  joint  a'c',  as  at  o',  whose  horizontal  projection 
is  o,  on  the  horizontal  projection,  os,  of  the  element  at  o". 

Considering  then  the  stone  below  the  joint  a'c',  a"c",  its 
radial  surfaces  are  si"ao  ;  oan,  where  on  —  o'c'  —  o"n"  is  an 
elliptic  arc  ;  and  ui'an. 

This  design  is  more  complex  than  the  usual  one,  but  gives  a 
greater  increase  of  radial  thickness  toward  the  springing  of  the 
elliptic  arch.  Indeed,  unless  the  radius  CjE'  exceeds  a  certain 
limit,  relative  to  CD'  and  D'E',  the  joint,  or  normal  e"F"  will 
be  less  than  D"E",  which  is  quite  undesirable,  relative  to  sta- 
bility. The  joints  of  the  intrados,  being  the  most  important, 
are  inked  full ;  that  is,  as  if  seen  from  below ;  as  it  is  often 
convenient  to  do  (8). 

The  outer  groin  may  be  elliptical,  found,  like  the  inner  one, 
from  given  extrados,  if  the  consequent  deviation  of  the  de- 
rived face  joints  from  a  normal  direction  be  only  very  slight. 

II.  The  Directing  Instruments.  —  These  can  be  constructed 
as  in  the  last  problem. 

III.  Application.  —  See  also  the  last  problem. 

Examples.  —  1°.  Let  both  arches  be  circular. 

2°.  Let  both  be  elliptical. 

3°.  Let  their  axes  intersect  at  any  other  than  a  right  angle. 

4°.  Turn  the  figure  right  for  left,  and  ink  the  plan  as  seen  from  above. 

5°.  Construct  the  patterns  for  the  key-stone  of  the  groin ;  and  of  any  other 
stone,  both  in  the  groined  and  in  the  cloistered  angle. 

6°.  Assume  an  elliptic  extrados  of  suitable  proportions  on  the  elliptic  arch,  and 
find  the  corresponding  curve  joining  the  corresponding  extremities  of  the  normal 
joints  of  the  circular  arch. 


STONE-CUTTING.  37 

Conical  or  Trumpet  Arches. 

58.  When  two  walls,  whether  having  vertical  or  sloping 
faces,  meet,  so  as  to  enclose  any  angle,  it  is  sometimes  desira- 
ble to  connect  them  at  or  near  the  top  by  self-supporting 
masonry,  which  may,  for  example,  afford  additional  area  for 
standing  or  passage.  The  original  ground  room  enclosed  by 
the  supposed  angle  at  which  the  walls  meet  is  not  encroached 
upon  by  the  arched  connection  of  the  walls,  since  the  latter  is 
self-supporting  from  the  walls. 

Such  arch  work,  projecting  from  one  or  more  wall  faces,  or 
piercing  them,  and  generally  with  a  conical  intrados,  has  been 
called  a,  trumpet. 

Peoblem  VIII. 

A  trumpet  in  the  angle  between  tioo  retaining  walls. 

I.  The  Projections.  —  1°.  Description.  Let  AC  and  BC,  PI. 
V.,  Fig.  36,  be  the  horizontal  traces  of  two  walls,  enclosing  a 
right  angle  ;  and  whose  faces,  CAtZF,  and  CBeF,  have  the  same 
slope  or  batter  of  2  to  5. 

Let  FDE  be  the  level  top  of  a  quadrantal  platform,  with  a 
radius  of  6' :  10"  ;  and  forming  a  quarter  of  the  base  of  an  in- 
verted oblique  cone,  whose  axis  coincides  with  FC  —  F'C,  the 
line  of  intersection  of  the  faces  of  the  walls  ;  and  a  portion  of 
whose  convex  surface  forms  the  front  face,  ADEB J  —  A'D'E' 
B'J',  partly  broken  away  on  the  right,  of  the  platform. 

This  platform  is  then  supported  by  a  trumpet,  whose  sur- 
face, AJBC  —  A' J'B'C,  is  a  segment  of  a  cone  of  revolution 
whose  axis  HC  —  C  is  perpendicular  to  the  plane  V,  and  in 
the  plane  H.  From  this  general  description,  we  proceed  to 
the  details  of  the  construction. 

2°.  Outlines  of  the  walls  and  platform.  —  Having  laid  down 
the  traces,  AC  and  BC,  of  the  walls,  CC,  the  horizontal  pro- 
jection of  their  intersection  will  bisect  ACB,  because  the  walls 
have  the  same  slope.  Next  draw  Cm,  and  on  it  lay  off  from 
C,  Jive  from  any  scale  of  equal  parts  ;  and  Ix,  parallel  to  BC  = 
two  from  the  same  scale ;  then,  by  the  given  conditions,  Cx 
will  be  the  section  of  the  face  of  the  wall  AdC,  made  by  the 
vertical  plane,  on  BC,  and  revolved  about  the  trace  BC,  into 


38  STEREOTOMY. 

H>  Now  lay  off  Cf=  the  height  of  the  platform,  and  Cm,  that 
of  the  wall,  draw  fW  and  mW,  parallel  to  Ix,  project  M'  at 
C,  and  F"  at  F,  on  CC,  and  draw  C'd  and  C'e  for  the  hori- 
zontal projections  of  the  upper  edges  of  the  walls,  and  FD  and 
FE,  respectively  parallel  to  them,  and  of  the  given  dimensions 
for  the  intersections  of  the  platform  with  the  faces  of  the  walls. 

3°.  Conical  front  of  the  platform,  and  its  vertex.  —  Sup- 
posing it  required  for  good  appearance  that  the  conical  front 
face  of  the  platform  shall  intersect  the  walls  in  their  lines 
of  declivity,  draw  DA  and  EB,  perpendicular  to  AC  and  BC, 
as  such  intersections  ;  then  AKB,  with  C  as  its  centre,  will  be 
the  horizontal  trace  of  this  conical  front :  and  as  DA  and  EB 
are  two  of  its  elements,  their  intersection,  V,  will  be  the  hori- 
zontal projection  of  its  vertex. 

The  projections  of  VD  and  VE  on  the  vertical  plane  BCM, 
are  BCM  and  B  ;  which  after  revolution,  as  in  (2°)  will  appear 
at  M'C  and  EB  produced,  which  meet  at  the  point  indicated  as 
V".  Then,  making  CV  =  BV",  we  have  V  the  vertical  pro- 
jection of  the  vertex  of  the  conical  front  of  the  platform. 

4°.  Outlines  and  joints  of  the  trumpet.  —  Let  the  vertical 
semicircle,  AHB  —  A'H'B',  on  the  chord,  AB,  of  the  horizon- 
tal trace  of  the  inverted  cone,  be  the  directrix  of  the  cone  of 
revolution  forming  the  trumpet.  Dividing  A'H'B'  conveniently 
into  equal  parts,  here  three,  project  i'  and  f,  the  points  of  di- 
vision, at  i  and  j,  giving  Ci —  Ci',  etc.,  as  elements,  and  joints, 
of  the  trumpet. 

To  find  the  face  line  of  the  trumpet,  find  where  its  elements 
meet  the  surface  of  the  cone  VV.  Since  the  vertex.  CC  of  the 
trumpet  cone  is  in  the  axis,  VC  — VC,  of  the  other  one,  this 
is  easily  done  by  assuming  any  element,  as  Ci  —  Ci',  of  the 
trumpet  and  finding  its  trace,  and  that  of  the  axis,  VF  —  VF', 
upon  the  plane  D'E'  of  the  upper  base  of  the  platform.  These 
traces  are  N'N  and  F'F,  giving  FN  as  the  trace  of  the  plane  of 
these  lines  upon  the  top  of  the  platform.  FN  cuts  the  circum- 
ference, EID,  of  the  platform  at  aa',  giving  aV  —  a'V  for  the 
element  of  the  cone  VV  in  the  same  plane  with  Ci  —  Ci1,  and 
hence  intersecting  the  latter  at  bb',  a  point  of  the  required 
face  line. 

The  highest  point.  —  This,  JJ7,  is  found  by  revolving  the  ele- 
ments VI  and  CH  of  the  two  cones,  and  in  their  common  ver- 
tical meridian  plane  VC,  about  any  convenient  vertical  axis,  as 


STONE-CUTTING.  39 

the  vertical  trace,  C'F',  of  that  plane,  till  they  fall  in,  or  par- 
allel to  the  plane  V^  Each  point  revolved  moves  in  a  hori- 
zontal arc,  at  its  proper  height,  as  can  be  read  from  the  figure, 
giving  V"K"I"  and  C"H"  as  the  respective  revolved  elements. 
These  intersect  at  J"  ;  which,  by  counter  revolution,  gives  JJ', 
the  required  highest  point. 

To  give  a  neater  design  to  the  top  of  the  platform,  the  planes 
of  the  joints  of  the  trumpet  radiate  from  the  axis  of  the  cone 
VV',  giving  the  radial  lines  as  aC  for  the  joints  of  the  platform. 
But  to  prevent  the  stones  from  coming  to  an  edge  along  that 
axis,  they  abut  against  a  conical  faced  stone  whose  upper  base 
is  gkh,  of  any  convenient  assumed  radius,  and  whose  intersec- 
tion ucv  —  u'c'v'  with  the  trumpet  cone  may  be  found  as 
A5B  —  AW  J'  was. 

This  stone  may  be  built  into  the  walls,  as  along  the  planes 
Qgu  and  hvV,  to  any  extent  desired  for  stability.  Also  the 
side  stones,  as  A'D'a'b',  of  the  trumpet  may  be  likewise  built 
into  the  wall  to  avoid  a  thin  edge  along  AC  —  A'C.  The 
small  component  of  the  reactions  of  the  side  stones  tending  to 
thrust  the  central  one  forward  would  be  sufficiently  resisted  by 
the  adhesion  of  the  cement.  Otherwise :  the  intermediate 
stones,  one  or  more,  could  easily  be  supported  by  forming  the 
stone  ghuv,  as  indicated  in  Fig.  37,  with  a  conical  step  as  s. 

II.  The  Directing  Instruments,  —  These,  taking  the  central 
stone  for  illustration,  will  consist  of  patterns  of  its  four  lateral 
faces,  with  a  few  bevels. 

After  Nos.  1  and  2,  there  will  be  No.  3,  the  pattern,  aknp, 
of  the  top,  No.  4,  that  of  the  two  equal  radial  beds,  and  No.  5, 
that  of  the  conical  intrados  ;  which  are  constructed  as  follows  : 
No.  4  shows  the  full  size,  and  real  form  of  the  radial  bed  npqr, 
which  is  supposed  to  be  revolved  about  its  horizontal  upper 
edge  np,  till  horizontal ;  when  r,  its  lowest  point,  will  be  found 
at  a  distance  s^,  Fig.  42,  from  np,  equal  to  the  hypothenuse 
of  a  right  angled  triangle,  of  which  rs,  perpendicular  to  np,  and 
the  vertical  distance  of  r'  below  n'p',  are  the  other  sides.  Also, 
WjSj  =  ns.  Finding  tyx  in  like  manner,  we  have  the  pattern 
No.  4,  which  will  serve  for  both  of  the  radial  beds  of  this 
stone. 

No.  5  is  the  development  of  the  conical  intrados,  made  by 
describing  the  arc  iyjx  =  i'j',  and  with  the  radius  C"H';,  =  the 
slant  height  to  the  circular  directrix  AB  —  A'H'B',  and  laying 


40  STEREOTOMY. 

off  the  true  lengths  of  the  elements  as  CA  =  O'b"'  =  the  true 
length  of  Cb —  Ob'  revolved  first  at  Ob"  into  the  vertical  plane 
COT'  and  thence  to  O'b"'. 

Bevels,  Nos.6  and  7,  set  to  the  angles  s^qi  and  s1n1r1^  and 
held  in  the  plane  CC'F',  will  be  useful  in  giving  the  positions 
of  the  two  ends.  The  latter  surfaces,  being  parts  of  oblique 
cones,  can  be  developed  only  by  the  usual  construction,  in  which 
the  intersection  of  such  a  cone  with  a  sphere  whose ,  centre  is 
the  vertex  (VV)  of  the  cone  is  found  ;  the  development  of 
such  intersection  being  a  circle.  But  these  developments  are 
unnecessary. 

Let  rs  now  be  considered  as  the  horizontal  trace  of  a  vertical 
plane,  perpendicular  to  the  edge  np,  and  cutting  the  adjacent 
radial  bed  in  sr,  and  the  top  in  a  line  also  horizontally  pro- 
jected in  sr.  By  revolving  the  former  sr  about  the  latter  till 
horizontal,  the  true  size  of  the  diedral  angle  between  the  top, 
and  the  radial  bed  npr,  will  be  found.  A  bevel,  No.  8,  set  to 
this  angle  will  be  useful. 

III.  Application.  —  Having  chosen  a  suitable  block,  in  which 
the  finished  stone  could  be  inscribed,  first  form  the  top,  by  No. 
1,  and  by  its  pattern  No.  3  ;  next,  the  radial  beds,  directing 
their  position  by  No.  8,  and  their  form  by  No.  4  ;  next  the  con- 
ical intrados,  of  which  No.  5  is  the  pattern.  All  the  edges  of 
the  two  ends  will  thus  be  known,  and  as  their  radial  edges  are 
elements  of  the  cones  to  which  they  belong,  it  is  only  necessary 
to  transfer  to  ap  and  bq,  from  the  drawings,  points  where  ele- 
ments meet  those  lines,  and  bring  the  front  end  apbq  to  its 
proper  conical  form  by  cutting  away  the  stone  till  No.  1  will 
apply  to  it,  at  the  corresponding  points  of  division  of  ap  and 
bg.  Hence  it  is,  that,  as  already  said,  the  tediously  found  de- 
velopments of  these  conical  ends  may  be  dispensed  with. 


Problem  IX. 

A  trumpet  arched  door  on  a  comer. 

I.  The  Projections.  1°.  Outlines  of  the  Plan.  —  PI.  V., 
Fig.  40.  The  projections  are  here  arranged  partly  with  a  view 
to  the  greatest  compactness.  Two  walls  of  an  enclosed  space, 
and  of  the  thickness,  AC  '==  BD  =  4'  :  9",  meet  at  right  angles. 


STONE-CUTTING.  41 

From  a,  the  equal  distances  aA  and  «B,  each  =  8'  :  6"  are  set 
off  as  the  external  limits  of  the  trumpet  as  seen  in  plan. 

If  CD,  the  width  of  the  door  were  also  given,  it  would  deter- 
mine the  angle  A/B  at  the  vertex  /  of  the  conical  surface  of 
the  trumpet.  We  here  suppose  A/B  =  90°,  which,  with  the 
previous  data,  makes  CD  =  5'  :  3|"  very  nearly.  A  vertical 
semicircle,  A.5B,  which,  revolved  about  AB  till  horizontal,  gives 
AEFB,  is  taken  as  the  base,  or  linear  directrix  of  the  trumpet. 
Then  CdD  is  a  vertical  semicircular  section  of  the  trumpet,  and 
also  of  a  second  conical  zone  CEFD  whose  vertex  is  e,  and 
which  serves  to  widen  the  approach,  EFH,  to  the  door.  To 
avoid  crowding  the  figure,  CD  is  shown  as  a  single  line,  which 
it  might,  in  fact,  be,  in  case  of  an  opening  having  no  gate,  or 
of  a  thin  iron  gate  ;  but  in  case  of  a  gate  of  considerable  thick- 
ness, the  edge  at  CdD  should  be  cut  away  giving  a  narrow  cyl- 
indrical band,  fitted  to  the  gate  top  ;  or  there  should  be  a  gate 
recess  as  in  PL  VI.,  Fig.  45. 

2°.  The  Elevation  in  general,  and  plans  of  the  elements.  — 
The  foregoing  being  the  main  features  as  seen  in  plan,  the  ele- 
vations are  shown  on  two  vertical  planes  ;  one  having  aB,  and 
the  other,  A"B"  for  its  ground  line.  Of  these,  only  the  former 
is  necessary  for  the  purposes  of  the  mason,  showing  as  it  does 
the  true  sizes  of  the  lines  in  the  external  face  of  one  of  the 
walls ;  while  the  other,  on  the  vertical  plane  at  A"B"  is  only 
useful  as  helping  to  give  an  idea  of  the  structure,  as  seen  in 
looking  directly  through  the  doorway,  in  the  direction  of. 

The  vertical  planes  Aa  and  Ba,  being  parallel  to  the  respec- 
tive opposite  elements,  B/  and  A/,  cut  the  trumpet  cone  AB/ 
in  equal  parabolas  ;  hence,  to  avoid  a  too  great  inequality  in 
the  sizes  of  the  arch  stones  as  seen  in  the  exterior  of  the  walls, 
divide  the  semicircle  A/B,  or  its  equal  A"b"F",  into  conven- 
ient unequal  parts,  the  largest,  W'l"  =  Bl!  being  laid  off  from 
the  springing  at  B"  of  the  arch.  Project  V,  or  I"  ;  h',  or  h"  ; 
etc.,  at  I,  h,  etc.,  and  through  I,  h,  etc.,  dxa,wflJc,  fg,  etc.,  hori- 
zontal projections  of  elements  of  the  trumpet.  These  at  n,  i, 
etc.,  pass  to  the  conical  zone  CEFD,  as  at  nm,  and  thence  to 
the  cylindrical  band  EGHF  in  parallel  elements,  as  mo. 

3°.  To  find  the  parabola,  Aa,  in  its  own  plane. 

1st.  Without  the  elevation  on  A"W.  —  The  trumpet,  being 
a  cone  of  revolution,  and  its  axis  af  horizontal,  its  elements  fa, 
fg,  etc.,  revolved  about  its  axis,/r,  and  towards  B,  will  come 


42  STEREOTOMY. 

to  coincide  with  the  extreme  element  /B  produced,  as  at  fax, 
fgv  etc. ;  where  aa\,  ggv  etc.,  perpendicular  to  fx,  are  the  hor- 
izontal projections  of  the  arcs  described  by  a,  g,  etc.  Then  a', 
g',  etc.,  vertical  projections  of  a,  g,  etc.,  extremities  of  elements 
of  the  trumpet,  are  at  the  intersections  of  the  perpendiculars 
aa',  gg' ,  etc.,  to  the  ground  line  aB,  with  the  arcs  aYa',  gig',  etc., 
all  having  x  for  their  centre,  and  which  being  in  the  vertical 
plane  on  «B,  are  seen  in  their  true  size.  (The  arc  kjc',  being 
confused  with  B&',  is  not  shown.) 

2d.  With  the  use  of  the  elevation  on  A"B".  —  Having  h",  for 
example,  vertical  projection  of  h,  draw  f'h",  the  vertical  pro- 
jection of  the  element  fh,  and  project  g  upon  it,  at  g".  Then 
gr  is  at  a  height  gg',  equal  to  that  of  g"  above  A"B"  ;  and  in 
like  manner  other  points  of  Ba',  except  a',  can  be  found.  As 
before,  aa'  =  aax. 

4°.  Determination  of  the  radial  beds.  —  These,  if  the  face 
joints,  as  g'V,  were  made  normal  to  the  parabolas,  of  which 
Ba'  is  one,  would  be  determined  by  these  joints,  with  the  ele- 
ment joints  fg,  etc.,  and  hence  could  not  also  contain  the  axis 
fx  of  the  trumpet,  since  g'V,  etc.,  if  normal  to  Ba',  do  not  in- 
tersect that  axis.  But  if  these  radial  beds  do  not  contain  the 
axis  fx,  which  is  also  the  axis  of  the  cone  cEF,  and  of  the  cyl- 
inder EGHF,  their  planes  cannot  cut  the  two  latter  surfaces  in 
elements,  and  the  stones  of  the  trumpet  would  properly  termi- 
nate in  a  vertical  plane  on  CD,  and  be  succeeded  by  others, 
radiating  from  the  axis  fex,  and  covering  the  surfaces  named 
between  CD  and  GH. 

We  therefore  choose  beds  radiating  from  the  common  axis 
fex  and  extending  from  Aa  and  Ba  to  LH.  The  top  edges  of 
these  beds,  in  the  horizontal  surfaces  as  I'K'  —  F'K"  will  then 
be  parallel  to  fx ;  and  the  face  joints  g'V  —  g"V,  etc.,  will 
radiate  from  the  point  x,  f"  in  the  plane  Ba. 

II.  The  Directing  Instruments.  —  These,  besides  Nos.  1  and 
2,  (7)  will  consist  of  patterns  of  the  surfaces  of  the  stones, 
with  such  other  bevels  besides  No.  2,  as  may  be  considered  use- 
ful as  checks. 

Taking  the  stone  gkinmor  —  g'VK'J'k'  —  I"K"J"n"i"r"m", 
No.  3,  the  pattern  of  its  back,  is  I"K"3"r"m",  which  is  in  the 
vertical  plane  LH. 

No.  4,  is  the  pattern  of  its  front,  VK'J'Jc'g'. 

Nos.  5  and  6,  are  the  patterns  of  the  radial  beds  on  f"Jrr, 


STONE-CUTTING.  43 

and  on /"I".  These  are  both  shown  as  revolved  about  the  axis 
fx  of  the  trumpet,  till  they  become  horizontal.  Thus  HI2  = 
I"r"  ;  gj.!  =  g'l'  ;  Dgi  shows  the  true  length  of  ig  —  i"g"  ;  etc. 

No.  7,  the  pattern  of  the  top,  is  a  trapezoid  of  altitude  I"K" 
and  bases  equal  to  JiJ2  and  LI2. 

No.  8,  is  a  pattern  of  the  conical  intrados,  gink,  found,  if 
flexible,  as  in  previous  similar  constructions  by  developing  the 
cone  whose  vertex  is  /.  But  as  it  is  only  the  elements  of  the 
intrados  that  must  be  found,  it  is  enough  to  develop  the  pyra- 
mid whose  edges  coincide  with  these  elements.  Hence  in  Fig. 
41,  the  chords  BZ  and  Ih  are  equal  to  the  chords  Wl"  and  l"h", 
Fig.  40  and  nikg  is  the  pattern  required. 

Flexible  patterns  9  and  10  of  inm,  and  srom  can  obviously 
be  found.  The  vertical  surface  on  K'J',  forming  No.  11,  is 
simply  a  rectangle,  =  K'J'  X  JiJ2- 

Nos,  12, 13,  and  14,  will  be  bevels  set  to  the  respective  angles 
K"l"g",  between  the  top  and  a  radial  bed ;  K"J"k"  ;  and  HFD, 
between  a  vertical  side  as  J'K'  and  the  front,  and  taken  in  a 
horizontal  plane. 

Fig.  44,  is  an  oblique  projection  of  the  stone  just  described, 
made  intelligible  by  means  of  the  like  letters  at  like  points. 

III.  The  Application.  —  First,  work  the  back  by  No.  1,  and 
mark  its  form  by  No.  3.  Second,  all  the  lateral  faces  adjacent 
to  the  back  are  made  square  with  it  by  No.  2.  Also,  the  top 
and  the  vertical  side  on  J'K'  are  at  right  angles.  Third,  the 
forms  of  the  faces  just  mentioned  can  then  be  marked  by  their 
patterns  (5-7),  10,  and  11 ;  and  the  bevels,  12  and  13,  can  be 
used  as  checks  on  their  position.  Fourth,  make  the  front 
square  with  the  top,  or  at  the  angle  HFD  with  the  vertical  side, 
holding  No.  14  perpendicular  to  the  edge  J'K'. 

Fifth,  mark  the  elements,  gi  and  nk  by  No.  9,  whence  inm 
can  be  wrought  by  No.  1,  placed  upon  corresponding  points  of 
division  of  in  and  ms  into  equal  parts. 

The  upper  joints  of  the  trumpet  stones  being  parallel  to  fx, 
there  will  be  some  three-cornered  stones  adjacent  to  them  in 
the  wall,  where  the  joints  are  parallel  to  AC  and  BD. 

Examples.  —  1°.  Let  the  walls  include  any  other  than  a  right  angle. 
2°.  Let  their  exterior  be  a  vertical  tangent  cylinder  from  A  to  B. 
3°.  Let  the  cone  be  other  than  right  angled  at  its  vertex  f. 
4°.  Let  FH  be  increased  till  LLj  shall  be  long  enough  to  embrace  the  back  ends 
of  all  the  trumpet  stones. 


44  STEREOTOMY. 

5°.  Let  LLi  be  a,  sloping  wall. 

6°.  Let  there  be  a  batter  to  the  exterior  faces  Aa  and  Ba,  of  the  wall. 

7°.  Let  there  be  no  opening  at  CD,  and  describe  with  an  oblique  or  isometric 
projection,  the  stone  GECDFH  — f,  necessary  to  fill  the  opening,  and  admit  the 
extension  of  the  trumpet  surface  to  its  vertex  f. 


PftOBLEM  X. 

An  arched  oblique  descent. 

I.  The  Projections.  — Various  conditions  may  give  occasion 
for  a  structure  of  this  kind.  Thus  it  might  lead  from  a  side 
walk  to  an  underground  railway  ;  or  from  a  hydraulic  canal  to 
a  turbine  wheel  pit ;  or  it  might  cover  an  arched  stairway  lead- 
ing to  an  arched  gallery. 

1°.  The  perpendicular  projections.  — ABCD,  PL  V,  Fig.  43,  is 
the  horizontal  projection  of  the  section  in  the  springing  plane. 
A  vertical  plane  on  AC  here  makes  an  angle  of  28°  with  a  ver- 
tical plane  on  Ax  A  perpendicular  to  AB.  The  line  CD  —  D' 
is  one  springing  line  of  the  intrados  of  a  semi-cylindrical  gallery, 
of  radius  MD  =  11'  :  6"  ;  the  perpendicular  length,  AA15 
of  the  horizontal  projection  is  7  ft. ;  and  AB  and  EF  are  re- 
spectively 15  ft.  and  9  ft. 

Two  principal  vertical  planes  of  projection  are  used  ;  V,  that 
of  the  head,  on  AB,  and  one  Vi  whose  ground  line  is  BD.  The 
line  AB  is  at  a  height,  BB1?  =  3' :  3"  above  CD,  and  hence, 
strictly,  the  diameter  AB  of  the  vertical  projection,  AI'B,  of  the 
head  should  be  a  line  A'B'  (not  shown)  parallel  to  AB  and 
3' :  3"  above  it.  But  to  condense  the  figure,  and  because  this 
position  of  AI'B  is  not  essential,  the  vertical  plane,  V,  of  the 
head  is  revolved  about  its  trace  on  the  springing  plane  BXDC, 
instead  of  about  its  horizontal  trace. 

2°.  The  oblique  projections.  —  The  projection  of  the  arch 
upon  the  vertical  plane,  Vn  on  BD,  might  be  made  in  the  usual 
way,  by  projecting  lines  perpendicular  to  that  plane.  But,  as 
may  be  seen  by  trial,  the  result  would  be  a  much  more  com- 
plicated, but  no  more  useful  figure  than  the  present  projection 
on  BD  ;  which  is  an  oblique  projection,  formed  by  projecting 
lines  parallel  to  AB. 

Thus,  since  the  plane  V  is  vertical,  By  perpendicular  to  BD, 
is  its  trace  on  the  vertical  plane  Vn  and  is  also  the  oblique  pro- 
jection of  the  head,  on  Vi-     Likewise,  the  quadrant  DV,  being 


STONE-CUTTING.  45 

the  revolved  semi-right  section  of  the  gallery  reached  by  the 
descent,  MXD  is  the  oblique  projection  of  its  horizontal  radius, 
MD,  and  similarly  S,  T,  etc.,  are  obliquely  projected  on  Vi  a^ 
Si,  T\,  etc.  Then  making  S^  =  Ss  ;  T^  =  Tt,  etc.,  Dt^  is 
an  arc  of  the  section  of  the  gallery  by  the  plane  Vi5  and  D/2, 
where  j"j2  is  parallel  to  BJD,  is  the  oblique  projection  of  the 
cylindrical  face  of  the  descent.  That  is,  ^f'j^D  is  the  oblique 
projection  of  the  outlines  of  the  descent. 

Laying  off  the  heights  of  the  several  points  of  the  semicir- 
cular face  above  AB,  from  Bx  on  Blc/",  and  drawing  lines  par- 
allel to  BiD  and  limited  by  D/2,  through  the  points  so  found, 
we  find  the  elements  and  edges  parallel  to  them,  of  the  descend- 
ing arch.  Thus,  BJ)"  =  BB'  by  drawing  Wb  parallel  to  BB2 
and  the  arc  bb"  with  centre  Bi ;  then  b"i2  is  parallel  to  BiD,  and 
all  the  other  parallels  to  BiD  are  similarly  found,  as  may  be 
seen  by  the  figure. 

3°.  The  right  section.  —  This  is  in  any  plane  perpendicular 
to  the  elements  of  the  arch.  To  condense  the  figure  (though 
at  the  expense  of  confusing  it  somewhat,  having  first  sought  to 
make  it  on  the  largest  scale  possible),  assume  XY,  perpendicu- 
lar to  BD,  and  XI1?  perpendicular  to  BtD,  as  the  traces  of  such 
a  plane.  Then  as  usual,  choose  auxiliary  planes  parallel  to 
the  axis  of  the  cylinder ;  here,  vertical  planes,  parallel  to  Vi« 
Each  of  these  will  contain  an  element  of  the  arch ;  and  a  line 
of  the  plane  YXIX,  whose  horizontal  trace  will  be  on  XY  and 
whose  vertical  projection  will  be  parallel  to  XIj.  Thus  the 
plane  I'PI",  cuts  from  the  arch  the  element  at  P" —  P  I",  and 
from  the  cutting  plane  YX1\  the  line  whose  horizontal  trace  is 
p"  (intersection  of  the  horizontal  traces  XY,  and  Pp")  and 
whose  vertical  projection  is  pp\  parallel  to  Xl\  through  p  the 
projection  of  p"  on  XD.  Having  found,  as  above,  b"i2  also  p'q', 
by  making  B^'  =  B^  =  BQ  =  PP',  as  shown  ;  p',  and  i',  the 
intersections  of  pp'  with  q'p'  and  b"i2  are  two  points  of  the  ob- 
lique projection  of  the  right  section.  Then  making  p"P"  =  pp' 
and  p"I"  =  pi1,  as  shown  by  revolving  p'  and  i1  to  px  and  I2, 
about  p  as  a  centre,  and  projecting  p^  and  I2  by  the  lines  p^P" 
and  LJ"  to  P"  and  I"  on  PP",  we  have  the  position  of  the  points 
p'  and  i'  when  revolved  about  XY  into  the  horizontal  plane. 

All  other  points,  both  of  the  oblique  projection  and  the  re- 
volved position  of  the  right  section,  are  similarly  found,  as  is 
sufficiently  indicated  by  the  lettering  of  other  points. 


46  STEREOTOMY. 

Making  Xe  =  Xe^,  and  drawing  e  Y  to  Y,  the  intersection  of 
XY  with  DC,  the  horizontal  trace  of  the  springing  plane  ;  eY 
is  the  revolved  position  of  the  intersection  of  the  plane,  YXI, 
of  right  section,  with  the  springing  plane  BiDY. 

4°.  Special  Constructions.  —  Ye  is  made  to  pass  through  F, 
in  order  to  compare  FO'E  and  FO'"E  more  nearly,  though  this 
is  not  necessary.  Ye  is  thus  placed  by  first  assuming  YXIX  at 
pleasure,  and  finding  the  corresponding  position  of  eY  ;  when, 
if  this  position  does  not  contain  F,  draw  a  parallel  to  it  that 
will,  viz.,  eY  as  on  the  figure,  which  will  meet  DC  at  that  posi- 
tion of  Y  whence  the  corresponding  desired  position  of  YX  can 
be  drawn. 

The  tangent  to  FO'"E,  parallel  to  AB.  Considering  AB  for 
a  moment  as  a  line  in  the  revolved  plane  of  right  section  and 
parallel  to  such  a  tangent,  its  horizontal  trace  would  be  y  ;  and 
its  vertical  trace  B2  would  be  found  by  making  XB2  =  XB. 
Then  making  the  height  Bb3  =  B262,  we  get  b3y,  its  projection 
on  V,  and  the  parallel  tangent  at  K'  gives  K'K  two  projections 
of  the  point  of  contact  of  the  required  tangent  from  which  K" 
the  point  of  contact  on  the  revolved  right  section,  of  a  tangent 
parallel  to  CD  is  found  as  in  (3°). 

II.  The  directing  Instruments.  —  Taking  the  springing  stone, 
FBB'I'R',  these  will  be  as  follows,  besides  Nos.  1  and  2.  No. 
3,  the  top,  is  a  parallelogram  of  width  B"I",  and  length  b"i2. 
Then  with  centre  B,  and  radius  b"i2  cut  DM  at  is,  and  Bz3  will 
be  the  position  of  b2i2  —  B',  after  revolving  till  horizontal,  about 
AB  as  an  axis,  since  i2  is  in  that  right  section  of  the  gallery 
whose  horizontal  projection  is  MD.  Then  ij$  and  BP  will  be 
two  sides  of  the  parallelogram,  No.  3. 

No.  4,  the  pattern  of  the  vertical  side  of  the  stone,  is  of  per- 
pendicular width  =  eB",  bottom  length  =  BJD,  and  top  length 
=  b"i2.  One  end  is  the  vertical  line  BB',  the  other  the  arc 
Dii  of  DV  (2°)  corresponding  to  Di2. 

No.  5  is  the  pattern  of  the  right  section  eB"I"R"F,  used  in 
one  method  of  working  the  stone. 

No.  6,  =  FB'BFR',  is  the  pattern  of  the  plane  end. 

No.  7,  the  pattern  of  the  opposite  cylindrical  end,  dif- 
fers from  No.  6,  in  that  BB'  would  be  replaced  by  the  develop- 
ment of  the  arc  Di4 ;  and  RR',  by  that  part  of  DV,  correspond- 
ing to  Dn2,  while  the  developed  joint  RT  would  be  curved,  as 


STONE-CUTTING.  47 

found  by  means  of  an  intermediate  point  u,  as  in  Prob.  V., 
etc. 

The  remaining  patterns  require  the  construction  of  other  de- 
velopments. Making  FOT'E",  Fig.  43  (taken  on  CD,  only 
to  bring  the  figure  within  the  plate),  equal  to  the  right  section 
FR"N"E,  Fig.  42  ;  note  that  the  real  distances,  estimated  on 
elements,  from  the  right  section,  e'o'f^  to  the  heads  of  the  arch, 
are  seen  on  the  oblique  projection.  Then  make  F"F,  Fig.  43, 
=/iBi,  Fig.  42,  and  in  like  manner  passing  from  one  figure  to 
the  other,  make  R"R  =  p'q<  ;  0"0  =  o'o'»  ;  N"N  =  w'Nl5  and 
E"E  =  e'B1 ;  and  FRNE  will  be  the  development  of  the  face 
line,  FO'E,  of  the  plane  end  of  the  arch. 

Next,  make  FH,  Fig.  43,  =  BXD,  Fig.  42  (i.  e.  F'H=/1D), 
and  likewise,  RRi  =  q'r2 ;  NNX  =  N^,  and  EG  =  BjD  ;  and 
HRjNiG,  will  be  the  development  of  the  face  line  of  the  cylin- 
drical end  of  the  descent.  Then  No.  8  =  FRRXH,  the  devel- 
oped intrados  of  the  stone  considered. 

Finally,  the  centre,  00^  of  the  plane  end,  to  which  its 
joints  radiate,  is  at  the  distance  O^  from  the  centre  Oi(0") 
of  the  right  section  ;  hence  in  Figs.  43  and  42,  respectively, 
make  0102,  at  this  distance,  01B1  from  the  right  section  F"E"  ; 
make  R02  =  R'O  ;  or  r02  =  R"0",  and  02RI  =  OR'I'.  Then, 
in  the  two  figures,  make  Ilx  =  b"i2,  and  O2o2  =  BJD,  and 
IiRiOg  will  be  the  developed  joint  on  the  cylindrical  end,  cor- 
responding to  R'I'O,  Fig.  42,  on  the  plane  end.  Hence  No.  9, 
=  RIIjRj,  is  the  pattern  of  the  radial  bed  on  RT. 

No.  10,  =  FBHD,  Fig.  43,  and  similarly  found,  is  the  pat- 
tern of  the  springing  surface  whose  horizontal  projection  is 
FBHD,  Fig.  42. 

Besides  these  patterns,  bevels,  Nos.  11  and  12,  set  to  the 
angles  B'T'O"  and  i2b"B  ,  respectively,  will  be  useful  in  one 
method  of  working  the  stone.  Also  No.  13,  giving  the  angle, 
BB"I",  between  the  side  and  top  of  the  stone,  in  a  plane  of 
right  section. 

III.  Application.  —  1°.  The  method  by  squaring.  —  Choose  a 
stone  on  which  a  right  section  can  be  formed,  exterior  to  the 
finished  stone,  by  No.  1,  and  No.  5  =  eB"FR"F.  Next,  make 
all  the  lateral  surfaces  square  with  this  right  section,  by  Nos.  1 
and  2,  and  mark  their  edges  by  their  patterns,  Nos.  3,  4,  8,  9,  and 
10.     This  operation  will  give  all  the  edges  of  both  ends,  which 


48  STEREOTOMY. 

can  thus  be  formed  by  cutting  away  the  rough  stone  on  the  ends 
down  to  them,  applying  No.  1,  in  a  direction  parallel  to  AB  on 
both  ends.  This  method  is  simple  and  accurate,  but  wasteful 
of  the  stone  between  the  actual  plane  end  and  the  exterior 
provisional  right  section,  and  of  the  labor  of  making  the  plane 
surface  of  this  right  section.  It  may,  however,  be  employed  in 
all  cases,  like  many  of  the  preceding,  where  the  actual  ends  are 
curved,  or  oblique  to  the  right  section. 

2°.  The  method  by  oblique  angled  bevels.  —  Choosing  a  block 
in  which  the  finished  stone  can  be  inscribed,  work  first  the  ver- 
tical side,  that  being  the  largest,  and  mark  its  edges  by  No.  4. 
Second,  work  the  top  square  with  the  former,  if  the  arm  of  the 
square  in  the  top  be  guided  by  a  small  plane  bevel,  laid  in  the 
top,  and  set  to  the  angle  %BF.  Otherwise,  use  the  level,  No.  13, 
held  perpendicularly  to  the  top  edge  Bi  (BD  —  b"i^).  Thence, 
finish  the  top,  by  No.  3.  Likewise  work  the  under  side,  and 
the  radial  bed  on  BT  —  B/'F,  the  latter  by  No.  11,  held  per- 
pendicular to  the  top  edge,  11^  Fig.  43.  Next  proceed  with 
the  plane  end,  using  No.  12  to  give  its  position  relative  to  the 
top.  From  the  finished  plane  end,  the  lengths  at  all  points 
being  known  from  the  side  elevation,  the  remaining  sides  and 
the  cylindrical  end  can  be  easily  and  accurately  wrought. 

Examples.  —  1°.  Make  the  side  elevation  in  perpendicular  projection. 

2°.  Let  the  arch  ascend  from  the  plane  end  to  the  gallery. 

3°.  Construct  the  indicated  pattern,  No.  7. 

4°.  Let  the  descent  be  direct,  BD  perpendicular  to  AB. 

5°.  Construct  the  patterns  for  the  key-stone. 

6°.  To  avoid  confusion,  take  X  to  the  right  of  B. 


CLASS  III. 

Structures  containing  Warped  Surfaces. 

59.  Warped  faced  wing  walls.  —  Suppose  that  the  inner 
faces,  as  bm —  b'm',  PL  I.,  Fig.  7,  instead  of  being  vertical,  were 
sloping,  but  in  such  a  way  that  the  lowest  lines  of  the  fronts 
of  the  walls  should  be,  as  seen  in  plan,  parallel  to  bm  and  np. 
Thus  let  them  be  as  at  h'h".  The  rate  of  slope  at  mh,  where 
the  wall  is  highest,  would  then  be  less  than  at  bh",  where  the 
wall  is  lowest.  The  face  of  the  wall  would  therefore  be  a 
warped  surface,  and  would  be  a  portion  of  a  hyperbolic  para- 
boloid ;  generated  either  by  hh",  moving  on  bh"  and  mh  so  as 
to  remain  horizontal ;  or  by  bh",  moving  on  bm  and  hh",  and 
parallel  to  the  vertical  plane  on  mn. 

Example.  —  Construct  the  case  just  described  in  a  large  figure,  with  an  aux- 
iliary elevation  showing  the  face  of  one  of  the  wing-walls ;  and  take  the  joints  to 
coincide  with  positions  of  hh"  and  bh". 

Problem  XI. 

The  recessed  Marseilles  Crate. 

I.  The  Projections.  —  1°.  The  problem  is  this.  Given  a 
straight  wall  in  which  is  a  recess  with  diverging  sides,  and  in 
the  recess  a  round  topped  portal,  closed  by  gates  of  like 
form ;  it  is  required  to  cover  the  top  of  the  recess  by  a  surface 
which  shall  be  agreeable,  easily  constructed,  and  practicable  in 
not  interfering  with  the  turning  of  the  gate.  Thus,  having  a 
vertical  straight  wall,  PI.  VI.,  Fig.  45,  bounded  in  thickness 
by  the  parallel  planes  AB  and  C"D  ;  and  in  which  is  the  pas- 
sage EF,  having  a  semicircular  top,  E'G'F',  and  covered  by 
gates  of  like  form,  fitted  to  the  recess  EFHI  —  H'E'G'G"F'P; 
it  is  required  to  cover  the  diverging  recess  or  embrasure  re- 
maining between  the  vertical  planes  HI  and  AB,  in  the  man- 
ner enunciated. 

It  will  be  agreeable  that  AH  should  be  not  less  than  HG, 
the  width  of  the  gate;  and  that  the  front  top  edge,  AB — 


50  STEREOTOMY. 

A'K'B',  of  the  recess  should  be  arched,  in  which  case  the 
vertical  height,  G"K',  and  the  radius,  AO  —  A'O",  should  be 
so  adjusted  that  A'  and  B'  shall  not  be  lower  than  G",  the 
highest  point  of  the  gate. 

So  much  being  fixed,  let  the  axis,  OY  —  O",  of  the  arch,  and 
the  face  lines,  H'G'T,  of  the  gate  recess,  and  A'K'B',  of  the 
embrasure,  be  the  three  given  directrices  of  a  warped  surface, 
generated  by  the  motion  of  a  straight  line  upon  them.  (Des. 
Geom.  251.) 

One  of  these  directrices,  OY — O",  being  straight,  any  de- 
sired elements  of  the  proposed  warped  surface  may  readily  be 
found  by  noting  the  points  in  which  any  plane  containing 
OY  —  O"  cuts  the  other  two  directrices.  Thus  00" A'  is  a 
plane  containing  O Y  —  O",  the  point  AA'  of  the  front  face  line, 
and  cutting  HI  —  H'G'T  at  L'L,  giving  ALY—  A'L'O"  for  an 
element  of  the  warped  surface. 

But  the  limited  directrices  limit  this  warped  surface  by 
the  elements  AL  —  A'L'  and  BM  —  B'M',  so  as  to  still  leave 
undetermined  the  surfaces  projected  in  ALH  and  BMI. 

On  extending  the  warped  surface  just  formed,  by  producing 
the  directrix  A'K'B',  it  will  generally  intersect  the  sides,  AH 
and  BI,  of  the  recess  in  curves,  which  would  prevent  the  full 
opening  of  the  gates. 

We  therefore  proceed  to  complete  the  proposed  top  of  the 
recess  by  means  of  warped  surfaces  having  the  two  direc- 
trices, OY — O",  and  H'G"I',  in  common  with  the  preceding 
warped  surface,  and  for  a  new  third  directrix  a  curve  through 
II'  and  BB',  so  formed  as  not  to  interfere  with  the  full  open- 
ing of  the  gate.  This  third  directrix  is  conveniently  shown, 
first,  in  its  real  form,  by  revolving  the  face,  BI,  to  a  position 
parallel  to  the  plane  V,  when  BB'  will  appear  at  GB'". 

2°.  Determining  conditions  of  the  new  third  directrix.  These 
are:  — 

1st.  That  it  should  enclose  I'G",  and  be  tangent  to  it  at  I'. 

2d.  That  it  should  also  contain  the  point  B'". 

$>d.  That  the  new  warped  surface  directed  by  it  should  be 
tangent  to  the  preceding  one  along  the  common  element,  MB 
—  M'B',  in  order  to  avoid  any  visible  edge  of  transition,  or 
break,  in  passing  this  common  limit  of  the  two  surfaces. 

The  last  condition  will  be  fulfilled  if  the  two  warped  sur- 
faces be  made  to  touch  each  other  at  any  three  points  of  their 


STONE-CUTTING.  51 

common  element,  YMB  —  0"M'B'.  But  this  they  evidently 
do  at  the  two  points,  YO",  and  MM',  since  there  the  linear 
directrices  are  the  same  for  both  surfaces. 

Let  BB'  be  the  third  point  of  YMB  —  0"M'B',  at  which  the 
two  warped  surfaces  shall  be  tangent.  For  this  purpose,  they 
must  there  have  a  common  tangent  plane.  Such  a  plane  will 
be  determined  by  two  tangent  lines  at  BB',  of  which  the  most 
convenient  are  YMB  —  0"M'B',  which  is  tangent  to  itself ; 
and  B'T',  the  tangent,  at  BB',  to  the  directrix  A'K'B'.  Now 
M'N',  parallel  to  B'T',  is  the  trace  of  this  tangent  plane  on  the 
plane  HI ;  and  N',  where  it  meets  the  intersection,  I  —  I'N', 
of  the  planes  HI  and  IB,  is  one  point  of  its  trace  on  the  plane 
IB.  But  BB'  is  another  point  of  the  same  trace,  which  is, 
therefore,  in  revolved  position,  N'B'". 

The  third  directrix  of  the  new  warped  surface  is  therefore,  as 
seen  in  the  revolved  position,  a  curve  which  shall  be  tangent 
at  I'  to  I'G",  and  at  B'"  to  N'B'". 

3°.  Choice  of  curves.  —  Preferring  a  natural,  to  an  artificial 
curve  for  the  directrix  now  determined,  we  may  attempt  an 
ellipse  having  either  I'O"  produced  for  its  transverse  axis;  or  a 
line  from  I',  parallel  to  N'B'"  for  a  diameter.  But  in  either 
case,  unless  its  radius  of  curvature  at  I'  be  not  less  than  0"P, 
it  will  intersect  I'G",  and  thus  be  impracticable.  Hence  the 
choice  must  generally  lie  between  a  curve  of  two  centres  com- 
posed of  a  part  of  I'G",  and  an  arc,  tangent  to  it  and  to  N'B" 
at  B'" ;  or  a  tangent  line  to  I'G"  from  B'",  with  the  portion  of 
I'B'"  from  I'  to  the  point  of  contact. 

Preferring  the  former,  draw  B'"P,  perpendicular  to  B'N', 
and  equal  to  I'O"  ;  draw  0"P,  and  a  perpendicular  to  it  at  its 
middle  point  will  meet  B"'P  produced  at  the  centre  of  the 
required  arc.  But  this  centre  will  generally  be  quite  remote, 
and  too  acutely  determined  for  accuracy;  hence  proceed  as 
follows.  The  contact,  </'",  of  the  two  arcs  may  be  found  by 
an  application  of  the  problem  :  To  draw  a  line  through  a  given 
point,  which  shall  pass  through  the  intersection  of  two  given 
lines.  Thus,  construct  any  triangle,  as  0"P1,  of  which  O" 
shall  be  one  vertex ;  the  two  others  being  on  B'"P  and  li,  the 
perpendicular  at  the  middle  of  0"P.  Then  draw  2,  3,  parallel 
to  OP ;  3,  4,  parallel  to  PI ;  and  4,  2,  parallel  to  0"1,  will 
meet  3,  2,  in  a  point,  2,  of  the  required  radius,  0"2,  which  lim- 
its the  arc  ~B'"g'"  at  g'".    Having  thus  found  g'",  we  can,  when, 


52  STEREOTOMY. 

as  in  practice,  making  the  drawings  on  a  very  large  scale,  find 
the  arc  W"g'"  by  points  as  in  (30). 

4°.  Test  for  Interference.  The  next  step  is  to  ascertain 
whether  the  surface,  generated  by  the  gate-top  in  opening,  in- 
terferes with  the  top  of  the  recess.  The  former  surface  is,  for 
the  left-hand  gate,  for  example,  a  portion  of  the  annular  torus 
generated  by  the  revolution  of  the  circle  of  radius  0"H'  around 
H  —  H'H"  as  an  axis.  In  such  a  torus,  the  gorge  or  interior 
opening  reduces  to  nothing,  and  the  portion  used  in  the  prob- 
lem is  generated  by  the  quadrant  H'G".  The  proposed  test  is 
made  by  taking  horizontal  planes,  as  Q'a',  and  finding  whether 
their  intersections  with  the  two  surfaces  intersect  within  or 
without  the  jamb  AH.  Thus  the  plane  Q'a'  cuts  the  gate 
torus  in  the  horizontal  circle  of  radius  HQ,  and  the  top  of  the 
recess  in  the  curve  a5Q,  found  by  projecting  a',  5',  etc.,  upon 
the  horizontal  projections,  7c,  LA,  etc.,  of  the  elements  which 
contain  them.  When  the  circles  as  Qc?,  described  by  points,  as 
QQ'  of  the  gate,  everywhere  intersect  the  curves,  like  Qba,  cut 
from  the  recess  roof  by  the  respective  planes  of  these  circles, 
outside  of  AH,  as  at  d,  no  interference  exists.  But  when,  as 
at  n,  the  curve  L/c  and  circle  \me  intersect  within  the  jambs, 
there  is  an  interference.  In  the  latter  case,  one  or  more  of  four 
means  may  be  used  to  remedy  the  difficulty  :  — 

1st.  To  increase  the  radius  K'O',  estimated  from  K/. 

2d.  To  raise  the  point  K',  without  increasing  the  radius 
K'O'. 

2>d.  If  the  interference  is  very  slight,  a  portion  of  the  recess 
roof  may  be  hewn  out  to  coincide  with  the  torus  generated  by 
H'G",  without  disfiguring  the  surface  by  abrupt  or  too  obvious 
changes  of  form. 

4th.  The  radius  of  the  gate-top  may  be  slightly  diminished. 

II.  The  Directing  Instruments.  The  construction  of  these 
will  be  best  illustrated  by  showing  patterns  of  all  the  devel- 
opable surfaces  of  the  most  irregular  one  of  the  voussoirs,  viz., 
that  between  the  radial  joints  q'o'  and  r'u'. 

Besides  the  straight  edge,  and  square,  and  the  rectangular 
patterns  of  the  top,  and  of  the  vertical  side,  at  u'v',  of  this 
stone,  Nos.  (1-4),  there  are,  No.  5,  a  pattern  of  the  back  end, 
q'o'v'u'r' ;  No.  6,  that  of  the  front  end,  shown  at  Wm'o'v'u't' ; 
No.  7,  that  of  the  radial  joint  at  r'u',  which  is  shown  by  re- 


STONE-CUTTING.  53 

volving  it  into  the  horizontal  plane  CjB",  after  supposing  the 
back,  CD,  of  the  wall  to  coincide  with  the  vertical  plane. 
Then  E'rx  =  rr"  ;  cxgx  =  the  perpendicular  distance  of  g  from 
CD  ;  T1t1  =  CC",  etc.  No.  8  shows,  in  like  manner,  the  joint 
in  the  plane  00" o  ;  and  No.  9  ==  B^ij,  Fig.  46,  the  true 
size  of  the  surface  \g —  B'g't',  forming  a  part  of  the  jamb  IB. 
Bevels  may  also  be  provided,  set  to  as  many  of  the  diedral 
angles,  as  m'o'v',  m'q'r',  etc.,  as  may  seem  best.  Nos.  10,  11, 
etc. 

III.  Application.  Form  the  plane  rectangular  top,  to  the 
dimensions,  o'v'  and  CC  of  pattern  No.  3 ;  then  the  three 
vertical  plane  surfaces,  viz.,  the  two  ends,  and  the  side  on  wV, 
square  with  each  other  and  with  the  top,  and  shaped  by  their 
patterns,  4,  5,  and  6.  Work  the  radial  beds  square  with  the 
back,  just  completed  by  No.  5,  checking  their  positions  by  the 
right  section  bevels,  10,  11,  etc. ;  and  scoring  their  edges  by 
patterns  7  and  8.  The  portion,  Wr-^sJ^  determines  the  cylin- 
drical surfaces  on  r'q'  and  p's',  and  the  annular  plane  portion, 
p'q'r's'. 

Every  edge  of  the  warped  portion  of  the  stone  being  now 
determined,  this  surface  can  be  wrought  by  the  straight  edge, 
No.  1,  held  in  the  direction  of  elements  of  the  surface,  and 
these  will  be  found  by  transferring  their  extremities  as  k'  and 
A',  M'  and  B',  from  the  drawings  to  the  stone. 


THE   OBLIQUE   ARCH. 

60.  This,  the  most  extended  of  all  the  problems  in  Stone 
Cutting,  is  usually  made  the  subject  of  a  separate  treatise  ;  for 
which  its  many  and  marked  varieties,  as  well  as  its  complexity, 
make  it  sufficient.  Yet,  by  the  full  and  careful  exhibition  of 
all  the  essential  features  of  its  usual  form,  the  student  can  be 
prepared  both  to  design  and  superintend  the  construction  of 
an  oblique  arch  as  commonly  built,  and  to  read  the  works  in 
which  the  subject  is  treated  more  elaborately. 

Preliminary  Topics. 
Elementary  Mechanics  of  the  Arch. 

61.  Let  ABC  be  an  ordinary  semi-cylindrical  arch,  of  which 
we  will  first  consider  only  one  half,  as  ABTa,  Fig.  4.     Let  G 


54 


STEEEOTOMY. 


represent  the  centre  of  gravity  of  this  half,  and  GH  its  weight, 
acting  vertically  downward.     The  half  ABT,  as  a  whole,  and 


-£■ 


Fig.  4. 

thus  actuated  by  its  unresisted  weight,  would  fall,  by  turning 
about  a  as  a  centre.  As  it  would  prevent  the  use  of  the  arch 
to  oppose  GH  by  props  underneath,  it  is  counteracted  by  a  hori- 
zontal force  acting,  as  at  T,  at  any  point  of  BT,  and  this  hor- 
izontal force  consists  in  the  reaction  of  the  other  half,  BTC,  of 
the  arch  and  its  immovable  backing.  This  understood,  as  a 
force  may  be  considered  as  acting  at  any  point  on  its  own 
proper  line  of  direction,  the  pressure  at  T,  and  the  weight  con- 
centrated at  G,  may  be  considered  as  both  acting  at  g  ;  the  for- 
mer at  gt,  the  latter  at  gh,  whence  gR,  would  be  their  resultant. 
Now  it  is  necessary  for  the  stability  of  the  half  arch  that  gH 
should  intersect  the  base  of  the  arch  between  A  and  a,  to  pre- 
vent rotation  about  one  or  the  other  of  those  points ;  and  what 
is  thus  evident  for  the  half  arch  as  a  whole,  is  true  of  the  sep- 
arate stones  of  which  it  is  composed.  That  is,  by  considering 
the  successive  segments  from  BT  to  the  successive  radial  joints 
of  the  arch,  in  the  same  manner  as  just  explained  for  the 
whole,  we  should  find  a  series  of  forces  like  gR,  one  for  each 
segment,  and  whose  intersections  would  form  a  polygon,  called 
the  line  of  resistance,  which  should  lie  wholly  within  the  arch 
in  order  to  secure  its  stability. 

62.  Passing  to  the  oblique  arch  (25),  Fig.  5,  it  is  evident 
from  the  foregoing  explanations  that  the  portions,  ABC  and 
ADEC,  of  the  left  half,  are  only  more  or  less  imperfectly  sup- 
ported by  the  opposite  half.  Hence,  in  discussing  the  oblique 
arch  relative  to  its  stability,  it  is  usual  to  consider  it  as  divided 


STONE-CUTTING. 


55 


by  planes  parallel  to  a  face,  as  GH,  into  an  indefinite  number 
of  laminse  ;  each  of  which  will  be  a  right  arch  of  a  span  equal 
to  ab,  the  oblique  span  of  the  given  oblique  arch. 

[c 


Fig.  5. 


That  is,  the  "  thrust"  "  lines  of  pressure"  or  "  lines  of  re- 
sistance "  in  an  oblique  arch,  are  assumed  to  act  in  planes 
parallel  to  its  faces. 

The  resulting  standard  or  essentially  perfect  design  for  an 
Oblique  Arch. 

63.  The  conclusion  of  the  last  topic  affects  the  form  and 
disposition  of  the  joints  of  an  oblique  arch,  and  thence  their 
graphical  construction,  in  the  following  manner :  — 

When  ttvo  surfaces  are  pressed  together,  the  pressure  at  each 
point  should  act  in  the  direction  of  the  normal  to  the  surfaces 
at  that  point.  Else  it  can  be  resolved  into  two  components : 
one,  normal  to  the  surfaces ;  and  one  parallel  to  them,  which 
will  tend  to  produce  slipping. 

Thus,  if  the  surfaces  of  two  bodies  meet  in  the  plane  whose 
trace  upon  the  paper  is  AB,  Fig.  6,  and  are  acted  upon  by  a 
force  producing  a  pressure  at  any  point,  p,  which  may  be  rep- 


56 


STEEEOTOMY. 


resented  by  OF=pTi1,  this  pressure  may  be  decomposed  at 
any  point  as  p  of  its  line  of  direction  into  the  normal  com- 


0 

s 

A 

V 

St 

A 

V 

\v 

Fig.  6. 



... 

*i    Pi 

ponent  pN1  =  ON ;   and  the  parallel  component  pS1  =  OS ; 
which  last  will  tend  to  produce  slipping  in  the  direction  pSv 

64.  In  applying  this  principle  to  the  design  of  the  joints 
seen  on  the  intrados  (19)  of  an  oblique  arch,  the  "  transverse" 
"heading"  or  "broken"  joints  are  made  in  planes  parallel  to 
the  faces,  and  thus  represent  on  the  arch  itself  the  direction  of 
its  thrust.  The  "  longitudinal"  "  coursing"  or  "  continuous  " 
joints  are  then  made  so  that  each  shall  intersect  at  right  angles 
all  the  transverse  joints  which  it  meets. 

65.  On  account  of  the  rectangular  intersections  of  the  joints, 
arches  thus  designed  are  often  described  as  forming  the  orthog- 
onal system.  On  account  of  the  equilibrium  of  the  pressures 
thus  acting  in  them,  they  are  also  often  called  equilibrated 
arches. 

The  coursing  joint  is  also  often  called  the  trajectory. 


Problem  XII. 

The  partial,  and  trial  construction  of  the  orthogonal  or  equili- 
brated arch. 

66.  Construction  of  a  coursing  joint.  —  1st.  In  vertical 
projection.  Let  the  plane  V?  PI-  "VI.,  Fig.  48,  be  considered 
as  parallel  to  the  faces  ABC  and  DEF  of  the  arch.  These 
semicircles,  with  the  equal  ones  having  any  convenient  number 
of  equidistant  points  <?,  ol5  o2,  etc.,  on  DC,  as  centres,  represent 
the  vertical  projections  of  sections  of  the  arch  parallel  to  its 
transverse  joints  (64).  ~Now,jirst,  a  line  is  normal  to  a  curve 
at  a  point,  when  it  is  perpendicular  to  the  tangent  at  that 
point ;  and,  second,  as  we  learn  from  descriptive  geometry,  if 


STONE-CUTTING.  57 

one  side  only  of  a  right  angle  be  parallel  to  a  plane,  the  projec- 
tion of  the  angle  on  that  plane  will  be  a  right  angle.  Hence 
in  the  figure,  as  the  tangents  to  the  circles  ABC,  etc.,  are 
parallel  to  V?  the  vertical  projection  of  the  normal  curve  to  all 
these  circles  will  be  perpendicular  to  the  projections  of  their 
tangents  at  its  intersections  with  these  circles.  Thus,  if  aQacx 
represent  an  arc  of  the  curve,  it  will  be  perpendicular  at  a  to 
the  tangent  aT,  and  therefore  tangent  at  a  to  the  radius  ao. 
Hence  we  have  the  following  construction  :  — 

Let  a  be  one  of  the  points  of  division  of  one  face,  ABC,  of 
the  arch,  through  which  a  coursing  joint  is  to  pass.  Draw  the 
radius  ao  ;  from  a\,  its  intersection  with  circle  ou  the  radius  a^  ; 
from  %,  intersection  of  axox  with  circle  o2,  draw  the  radius  a2o2 ; 
from  a3,  similarly  found,  the  radius  a3o3,  etc. ;  and  the  line 
aaxa2  .  .  . .  «n,  thus  found,  will  nearly  coincide  with  the  required 
curve ;  only  that  all  the  points  except  a  are  evidently  a  little 
too  low.  It  is  an  advantage,  however,  that  the  errors  are  wholly 
on  one  side  of  the  truth. 

Now  draw  aoj,  meeting  circle  ox  at  e1 ;  thence  oxo^  likewise 

giving  c2 ;  thence  c2o3,  etc.,  and  again,  acxe^ cn  will  be 

nearly  the  required  curve,  but  all  its  points  besides  a  will  be  a 
little  too  high.  How  now  shall  we  approximate  more  closely 
to  the  correct  curve  ? 

67.  At  this  point  a  general  observation  upon  the  relative 
exactness  of  constructions  made  by  plotting  to  scale  the  results 
given  by  analysis  from  numerical  data,  or  by  geometrical  con- 
struction from  graphical  data,  may  be  useful.  The  finest  lines, 
as  perfect  both  in  length  and  direction  as  can  be  laid  down  on 
a  large  scale,  and  with  the  nicest  care,  on  firm  and  smooth 
paper,  will  give  results  equal,  in  nearly  or  quite  every  case,  for 
all  practical  purposes,  to  those  of  analysis. 

The  perfect  length  and  location  of  lines  are  analogous  to 
absence  of  errors  of  calculation  ;  and  fineness  and  large  scale  of 
construction  are  analogous  to  the  carrying  out  of  numerical 
computations  to  a  large  number  of  decimal  places. 

Besides,  in  all  cases  where  drawings  are  necessary  as  guides 
in  the  execution  of  a  work,  the  results  of  calculation  are  ex- 
posed to  instrumental  errors  in  plotting  them  to  scale. 

68.  Returning  now  to  the  means  of  approximating  to  the 
true  curve,  any  one  or  more  of  the  following  methods  may  be 
employed. 


58  STEREOTOMY. 

1°.  Increase  to  any  desired  extent  the  number  of  circles  be- 
tween the  faces,  ABC  and  DEF,  of  the  arch ;  since  this  will 
evidently  cause  the  curves  aa5  and  ac5,  which  are  on  opposite 
sides  of  the  true  curve,  to  approach  each  other  ;  that  is,  will 
cause  either  of  them  to  approach  the  true  curve. 

2°.  Bisect  the  spaces  a±c^  a2c2,  etc.,  and  the  curve  through 
the  points  thus  formed  will  sensibly  coincide  with  the  true 
curve  ;  especially  in  a  figure  drawn  on  a  large  scale,- and  with 
numerous  auxiliary  circles. 

3°.  Bisect  the  distances  oox ;  o2o2,  etc.,  between  the  centres 
of  the  successive  circles  ABC,  etc.,  at  the  points  1,  2,  3,  etc., 
and  draw  al,  which  will  evidently  divide  alcl  at  some  point  as 
%,  not  shown.  Then  draw  %,  2 ;  meeting  a2c2,  at  a  point  n^, 
etc.,  and  a  %  n2,  etc,  will  very  nearly  coincide  with  the  true 
curve. 

4°.  Suppose  the  point  wx,  just  described,  to  be  a  point  of  the 
true  curve.  Then,  observing  that  the  curvature  of  the  required 
curve  rapidly  increases  as  it  ascends,  tangents  to  it  at  a  and  % 
would  meet  nearer  to  a  than  to  n.  Hence  assuming  t  as  one 
point  of  a  required  curve,  draw  to5,  and  take  r  on  the  part  ts 
and  a  little  nearer  t  than  to  s  ;  and  draw  ro±,  and  on  the  portion 
t^i  of  this  line  take  rx,  nearer  the  middle  of  t^  than  r  was  to 
the  middle  of  ts  ;  thence  draw  rxo3,  and  on  the  portion  t2s2  of 
this  line  take  r2  nearer  the  middle  of  t2s2  than  rx  was  to  the 
middle  of  t\%\ ;  and  so  on ;  and  tlfa,  etc.,  will  be  very  exactly 
the  true  curve,  tangent  at  t,  tx,  t2,  etc.,  to  io5,  ro4,  rxo%,  etc. 

Example. — Work  out  each  of  the  above  four  methods  with  the  circles  ABC, 
etc.,  drawn  to  a  radius  of  at  least  three  inches. 

69.  Horizontal  projection  of  a  coursing  joint.  —  Assum- 
ing merely  for  illustration  that  ac5  is  the  true  vertical  pro- 
jection of  such  a  joint,  simply  project  its  points  upon  the 
horizontal  projections  of  corresponding  circles,  as  a  at  a' ;  C\  at 
c[;  c2  at  c2,  etc. ;  where  the  accents  are  given  to  the  horizontal 
projection,  as  is  sometimes  done,  when  the  vertical  projection  is 
first  made,  or  of  most  service. 

70.  Development  of  a  coursing  joint.  —  Let  the  cylin- 
der A'C'D'F'  be  first  made  tangent  to  the  horizontal  plane 
along  the  element,  CF  —  C'F',  and  then  rolled  out  upon  that 
plane.  All  its  elements  will  then  be  parallel  to  C'F'  in  de- 
velopment, and  at  distances  from  it  equal  to  the  true  lengths 


STONE-CUTTING.  59 

of  the  arcs  of  right  section  between  them  and  C'F'.  Hence 
make  m"n'  equal  to  the  true  length  of  the  elliptic  arc  m'n' — • 
mn  of  right  section,  as  it  would  appear  projected  upon  a 
plane  perpendicular  to  the  axis  005  —  ooy  (This  construction 
is  not  shown,  for  want  of  room).  Likewise  v'a$  equals  the 
true  length  of  the  elliptic  arc  (arc  of  right  section,  whatever  it 
may  be,  in  any  case)  v'a'o — va0 ;  and  q'a"  equals  the  true 
length  of  a'q' — aq;  etc.  for  any  sufficient  number  of  points. 
Then  p'm"a'o;  C'a" ;  q'c'{,  etc.,  are  the  developments,  all 
alike,  of  the  equal  semicircles,  a'0p'  —  a0p  ;  OC' —  oC  ;  etc. ; 
and  a§avc'{c'i,  etc.,  is  the  development  of  the  coursing  joint 
a'0c5 —  a0c5;  and  it  crosses  the  developed  semicircles  at  right 
angles. 

71.  Identity  of  form  of  the  coursing  joints.  —  Through  any 
point,  as  a,  Fig.  48,  draw  an  element  meeting  the  other  circles 
at  points  as  5,  homologous  with  a.  It  is  perfectly  evident  from 
(70)  that,  if  we  construct  the  trajectory  passing  through  b,  as 
in  (6Q~),  it  will  be  of  the  same  form  as  that  through  a.  That  is, 
all  the  trajectories  have  parallel  tangents  at  points  on  the  same 
element,  and  hence  are  all  alike. 

Likewise,  the  horizontal  projections,  and  the  developments 
of  like  portions  of  the  trajectories  are  identical  in  form.  Hence 
having  any  projection  or  development  of  any  trajectory,  the 
portion  of  any  other  one  included  between  the  same  elements 
of  the  cylinder  would  be  drawn  simply  by  a  card-board  pattern 
of  this  initial  one. 

72.  Convergence  of  the  coursing  joints.  —  It  will  be  imme- 
diately evident  on  constructing  any  other  coursing  joint,  as  the 
one  through  m'm  for  example,  that  these  joints  converge,  rap- 
idly at  first ;  as  they  approach  CF  —  C'F'  in  the  direction  from 
C  to  F',  and  DA  —  D'A'  in  the  direction  from  D'  to  A'  when 
produced  indefinitely  in  both  of  these  directions. 

The  result  of  this  convergence  is,  that  no  two  stones  in  the 
same  course  are  alike  ;  though  stones  in  like  positions  in  differ- 
ent courses  are  so.  This  greatly  increases  the  number  of 
necessary  patterns,  difficulty  of  execution,  and  the  expense  of 
constructing  an  equilibrated  arch. 

73.  Creneration  of  the  joint  surfaces.  —  We  know  that  1°,  the 
normal  to  a  surface  at  any  point  isperpendicular  to  the  tangent 
plane  at  that  point ;  2°,  that  when  a  line  is  perpendicular  to  a 
plane,  the  projections  of   the  line   are  perpendicular  to  the 


60  STEREOTOMY. 

traces  of  the  plane.  Now  aT  is  the  vertical  trace  of  the  tan- 
gent plane  to  the  intrados  along  the  element  ab,  at  aa! ;  its 
horizontal  trace  would  be  a  line  through  T,  parallel  to  C'F'. 
Hence  the  normal  to  the  intrados  at  aa',  for  example,  is  oa  — 
a'a".  If  then  an  arm  de,  normal,  at  c2,  be  fixed  to  a  rod 
OxO  which  coincides  with  the  axis  005,  and  if  the  rod  and  arm 
move  together,  the  rod  moving  so  as  to  coincide  with  the  axis, 
and  the  arm,  so  that  the  point  c2  of  the  arm,  shall  continue  on 
the  semicircle  02u,  the  point  d,  considered  as  one  point  of  the 
extrados,  will  generate  the  transverse  'joint  of  the  extrados,  cor- 
responding to  the  semicircle  02u  of  the  intrados ;  and  the  por- 
tion c2d  will  generate  one  of  the  transverse  joint  or  heading 
surfaces ;  which  is  thus  everywhere  normal  to  the  intrados. 

Likewise,  the  rod  OxO  continuing  to  coincide  with  the  axis 
as  it  moves,  with  the  arm,  let  the  pair  move  so  that  the  point 
c2  of  the  arm  shall  trace  the  coursing  joint  a'c5,  the  point  d 
will  then  trace  the  extradosal  trajectory  corresponding  to  a'cs 
of  the  intrados,  and  the  line  c\d  will  generate  the  normal  joint 
surface  having  d'c5  for  its  directrix. 

74.  Nature  of  the  joint  surfaces.  —  These  are  evidently  both 
warped,  but  of  a  variable  twist ;  the  former,  the  transverse 
one,  is  so  by  reason  of  the  variable  relative  velocity  of  the  axial 
and  the  rotary  motions  of  the  frame  Oxed,  which  is  imme- 
diately evident  on  constructing  any  three  of  its  positions  ;  the 
latter,  the  coursing  joint  surface,  is  variably  warped  by  reason 
of  the  variable  curvature  of  the  coursing  joints. 

75.  Summary.  —  Without  detailed  drawings,  or  further  in- 
vestigation, five  grave  practical  objections  to  the  equilibrated 
arch  are  already  apparent. 

1°.  The  inequality  of  the  stones  in  each  course,  resulting  in 
the  evils  already  noted  (72). 

2°.  The  variable  twist  of  the  normal  joint  surfaces,  which 
still  further  complicates  their  execution. 

3°.  The  non-parallelism  of  opposite  faces  of  the  same  stone, 
whereby  the  actions  and  reactions  upon  those  faces  are  not  in 
the  same  line,  and  therefore  in  the  aggregate  yield  resultant 
couples,  tending  to  produce  rotation  of  the  arch  about  a  verti- 
cal axis. 

4°.  The  convergence  of  the  coursing  joints,  while  like  points 
of  each  are  not,  as  we  have  seen,  in  the  same  section  parallel 
to  the  face,  also  makes  the  stones  in  the  face  of  unequal  width, 


STONE-CUTTING.  61 

unless  they  are  arbitrarily  made  equal  by  breaking  joints  with 
those  behind  them. 

5°.  The  same  convergence,  finally,  renders  it  impossible  to 
build  equilibrated  arches  of  common  brick,  as  is  often  de- 
sirable. 

For  these  reasons,  we  shall  here  dismiss  the  equilibrated 
arch,  only  referring  the  reader,  who  wishes  to  become  familiar 
with  its  details,  to  the  works  of  Bashforth,  Grraeff,  and  Ad- 
HEMAR ;  and  shall  seek  an  approximation,  which,  as  nearly  as 
possible,  retains  the  merits,  but  avoids  the  defects  of  the  equili- 
brated arch. 

We  will  begin  by  seeking  a  simpler  form  of  warped  surface, 
normal  to  the  cylindrical  intrados,  for  the  heads  and  beds  of 
the  voussoirs. 

The  Helix. 

i 

76.  Let  o—  0',  12',  PL  VII.,  Fig.  50,  be  a  fixed  axis,  and 
let  0,0'  be  the  initial  position  of  a  generating  point  which  has 
these  two  simultaneous  motions ;  one  of  revolution  around  that 
axis,  and  one  of  translation,  parallel  to  it.  By  the  first  motion 
alone,  the  point,  00',  would  generate  the  circle  0,3,6,9,12  ;  by 
the  second  alone,  the  straight  line,  0  — -  0'12'  ;  but,  by  both 
motions  combined,  it  would  generate  the  curve  called  a  helix. 
The  height  0'12'  corresponding  to  a  full  revolution,  0,3,9,12,  is 
called  the  pitch  of  the  helix. 

The  usual  case  is,  that  each  of  these  motions  is  uniform, 
giving  rise  to  the  common  helix ;  or  simply  the  helix,  as  com- 
monly understood. 

77.  Three  elementary  properties  immediately  follow  from  the 
definition  just  given. 

1°.  The  helix  will  lie  on  the  convex  surface  of  a  cylinder  of 
revolution  whose  axis,  o —  0'12',  is  that  of  the  helix;  and  whose 
radius  is  the  perpendicular  distance  Qo  of  the  generatrix  00' 
from  the  axis. 

2°.  Since  the  two  component  motions  of  the  generatrix,  one 
around  the  cylinder  at  right  angles  to  its  elements,  and  the 
other,  parallel  to  them,  are  each  uniform,  the  path  of  this  point, 
i.  e.  the  helix,  must  cross  all  the  elements  at  a  constant  angle. 

3°.  It  now  follows  that  when  the  convex  surface  of  the  cyl 
inder  is  developed,  the  development  of  the  helix  ivill  be  straight .. 
for  the  elements  of  the  cylinder  will  be  parallel  in  development 


62  STEREOTOMY. 

and  a  line  which  crosses  parallels  at  a  constant  angle  must  be 
straight. 

It  thus  appears  that  the  joints,  CD15  KQ1?  etc.,  BJV,  C5, 
etc.,  Fig.  49,  primarily  chosen  as  convenient  and  sufficient  sub- 
stitutes for  the  theoretic  ones  of  the  orthogonal  system,  are 
helices. 

78.  The  construction  of  the  helix  also  follows  immediately 
from  its  definition  (76),  as  is  shown  in  PL  VIL,  Fig.  50. 

For  let  the  distance  0'12'  on  the  axis,  be  the  height  attained 
by  the  generatrix  while  also  making  one  revolution  around  the 
axis  from  0,0'  to  12,12'.  This  revolution  is  evidently  indicated 
in  horizontal  projection  by  the  circle  of  radius  o0.  Then,  as 
both  motions  are  uniform,  divide  this  circle,  and  the  correspond- 
ing height  (pitch)  0'12',  into  the  same  number  of  equal  parts, 
and  draw  horizontal  lines  through  the  points  of  division  on  the 
axis,  and  number  them,  as  shown,  to  correspond  with  the  num- 
bers of  their  horizontal  projections,  oO,  ol,  etc.  Then  project 
up  1,  2,  3,  etc.,  from  the  plan  upon  the  lines  similarly  num- 
bered in  elevation,  at  1',  2',  3',  etc.,  and  the  curve  0',  1',  2',  3', 
etc.,  will  be  the  vertical  projection  of  the  helix. 

Theorem  II. 

The  projection  of  the  helix  on  a  plane  parallel  to  its  axis  is  a 

sinusoid. 

A  sinusoid  is  a  curve  in  which  if  the  abscissas,  0'b',  0'g',  etc., 
are  equal  to,  or  proportional  to  the  arcs  of  a  given  circle,  the 
corresponding  ordinates  W,  2!g',  etc.,  are  equal  to  the  sines  of 
those  arcs.  Now  Vb'  =  the  sine  of  the  arc  01  in  the  plan  ; 
2'g',  =  that  of  the  arc  02  in  the  plan,  etc. ;  and  the  spaces  0'5', 
0'g',  etc.,  are  proportional  to  those  arcs.  Hence,  the  curve  0', 
1',  2',  3', 12'  is  a  sinusoid. 

The  Helicoid. 

79.  Let  now  all  the  points  of  the  horizontal  line  o0  —  0', 
PI.  VIL,  Fig.  50,  have  the  same  two  simultaneous  motions  that 
have  just  been  given  to  its  outer  point,  00'.  Every  point  of  the 
line  oO  —  0'  thus  describes  a  helix,  having  the  same  pitch,  0'12\ 
and  whose  horizontal  projection  would  be  a  circle  centred  at  o. 
All  the  consecutive  helices,  from  the  axis  0'12',  to  0',  1',  2', 


STONE-CUTTING.  G3 

3' 12',  and  so  on  outward  without  limit,  thus  described, 

will  constitute  the  surface  called  a  right  helicoid.  The  portion 
within  the  cylinder  of  radius  oO,  is  shown  by  the  shaded  area 
of  Fig.  50. 

80.  The  surface  just  denned  is  called  a  helicoid,  because  its 
generatrix,  the   line,  oO  —  0',  moves   upon  a  given  helix,  as 

0',  1',  2',  3' 12',  as  a  directrix ;  and  a  right  helicoid 

because  this  generatrix  is  perpendicular  to  the  axis  o  —  0'12'. 
The  surface  is  the  same  as  that  of  the  screw  surfaces  of  a 
square  threaded  screw,  or  of  the  plastering  on  the  underside 
of  circular  stairs. 

The  right  helicoid  is  evidently  normal  to  the  convex  sur- 
face of  the  right  cylinder  having  the  same  axis,  since  all  its 
elements  are  perpendicular  to  the  axis  of  that  cylinder. 

Thus  we  see  that  the  joint  surfaces  of  the  voussoirs  (18)  nor- 
mal to  the  cylindrical  intrados  of  the  oblique  arch  are  right 
helicoids. 

Problem   XIII. 

A  segmental  oblique  arch,  on  the  helicoidal  system. 

I.  81.  The  Projections.  Primary  outlines.  —  Let  ABCD, 
PL  VII.,  Fig.  49,  be  the  plan  of  the  intrados  of  a  segmental 
(27)  arch  having  a  square  span,  AG,  of  13  ft.  ;  an  angle  of 
skew,  CAB,  of  54°  ;  and  a  perpendicular  length,  yyx,  of  14 
feet,  between  the  vertical  planes,  AB  and  CD,  of  the  faces. 

Let  EF  —  O"  be  the  projections  of  the  axis  of  the  arch, 
taken  perpendicular  to  the  plane  V,  and  let  the  arc  A'E'B'  of 
120°,  centred  at  O",  indicate  the  extent  of  the  segment  of  the 
cylinder  of  revolution,  which  forms  the  intrados. 

Then  AB^C  is  evidently  (see  PI.  III.,  Fig.  28,  etc.,  the 
development  of  the  intrados  ;  where  CCX  =  15'. 708  =  the 
length  of  the  right  section  Cc3  —  A'E'B' ;  and  the  curves  ABX 
and  CT>1  are  the  developments  of  the  face  lines,  AB  —  A'E'B', 
and  CD  —  A'E'B'  ;  as  will  be  presently  explained  more  in 
detail. 

This  being  understood,  and  the  curve  CDj  representing,  at 
each  point,  the  direction  of  the  thrust  of  the  arch  at  that 
point  (62)  the  straight  line  CDX  symmetrical  with  the  curve,  so 
nearly  coincides  with  it  as  to  sufficiently  replace  it  as  a  proper 
direction  for  the  developed  transverse  joints  of  the  intrados ; 


64  STEREOTOMY. 

but,  being  straight,  is  evidently  the  development  of  a  helical 
arc  (77,  3°). 

This  settled,  next  divide  the  straight  line  CDj  into  the  odd 
number  of  equal  parts,  chosen  as  the  number  of  voussoirs,  in 
the  width  of  the  arch,  in  this  case  nine.  Then,  as  the  trans- 
verse and  coursing  joints  should  be  nearly  or  quite  at  right 
angles  to  each  other,  let  the  developed  coursing  joints,  B^v, 
etc.,  be  perpendicular  to  CD1?  or  as  nearly  so  as  the  length 
DjBj,  or  width  CCi,  if  unalterable,  will  allow.  Such  coursing 
joints,  being  evidently  both  parallel  and  equidistant,  the  arch 
may  be  built  of  brick  if  desired,  (75,  5°)  and,  if  of  stone,  all 
the  voussoirs  in  all  the  courses  will  be  of  equal  width  (75, 
1°,  3°,  4°.) 

Returning  to  Fig.  50,  and  comparing  Figs.  49  and  50,  the 
axis  of  the  arch  replaces  the  axis  o  —  0',  12',  and  is  perpendic- 
ular to  V-  Hence,  A'E'B'  is  the  vertical  projection  of  the 
right  cylindrical  surface  of  the  intrados,  and  of  all  the  helices 
traced  upon  it ;  and  the  horizontal  projections  of  these  helices 
will  be  similar,  in  form  and  construction,  to  the  vertical  projec- 
tion of  the  helix  in  Fig.  50. 

82.  Horizontal  projections  of  transverse  helical  joints  on  the 
intrados.  —  KQX,  parallel  to  CDX,  being  the  development  of 
one  of  these  joints  (77,  3°),  and  K&4  that  of  the  right  section 
at  K,  shows  that  Qx&4  is  the  partial  pitch  due  to  the  arc  A'E'B' 
of  the  circular  projection  of  the  same  helical  joint.  By  counter 
development,  Q:^4  returns  to  Qks.  Hence,  if  we  divide  A'E'B' 
and  Q&3,  each  into  the  same  number  of  equal  parts,  numbered 
similarly  from  B'  and  Q  as  zero  points,  the  points  of  the  hori- 
zontal projection,  KPQ,  of  the  helix  considered,  will  be  a,t  the 
intersection  of  the  projecting  lines  from  1',  2',  3',  etc.,  on 
A'E'B',  with  parallels  to  Kk3,  through  the  corresponding  points 
on  Q&3,  as  shown  in  the  figure.  Thus,  M2and  Bt  are  projected 
from  2'  and  3'  upon  the  second  and  third  parallels  from  Q. 

83.  Horizontal  projections  of  intrados al  helical  coursing  joints. 
—  In  like  manner,  RAT,  one  of  the  parallels  drawn  through 
points  of  equal  division  on  CDX  is  the  development  of  so  much  of 
coursing  joint  as  lies  on  the  given  segment  of  the  cylinder  taken 
to  form  the  arch.  Tr5  is,  therefore,  its  proportion  of  the  pitch 
(76)  of  a  coursing  joint,  and  A'E'B'  is  its  vertical  projection. 
Then,  as  before,  divide  Tr5  and  A'E'B'  into  the  same  number  of 
equal  parts,  number  similarly  from  B'  and  r5  as  zero  points, 


STONE-CUTTING.  65 

and  the  intersections  of  projecting  lines  from  1',  2',  3',  etc., 
with  the  parallels  to  Rrfi  through  the  like  numbers  on  Trs,  will 
be  points  R  .  .  .  R2,  R3,  etc.  of  the  coursing  helix  RST. 

84.  Construction  of  horizontal  projections  of  helices  from  their 
developments.  —  The  above  constructions,  having  been  inserted 
to  render  them  more  intelligible  by  their  analogy  with  Fig.  50, 
it  will  now  be  shown  that  the  sinusoid  (Theor.  II.)  projection 
of  the  helix  can  be  found  from  its  circular  projection  and  devel- 
opment. For  the  same  parallels  that  divide  Q^4  =  Q&3,  the 
pitch  of  KPQ  —  A'E'B'  into  equal  parts,  divide  the  develop- 
ment, KQ15  of  the  same  helix  into  the  same  number  of  equal 
parts  ;  and  the  like  is  true  of  ^R{£.  Hence,  in  cases  like  the  pres- 
ent, where  the  developments  of  the  helices  of  the  intrados  are 
given,  their  horizontal  projections,  as  QPK,  are  most  naturally 
found  as  the  intersections  of  the  projecting  lines  from  1',  2', 
3',  etc.,  on  A'E'B',  with  perpendiculars  to  the  axis  EF,  from  the 
developments  of  the  same  points,  04,  M3,  B4,  etc.,  thus  giving 
respectively  03,  M2,  B3,  etc.,  on  QPK. 

All  the  other  intradosal  helices  are  found  in  a  precisely 
similar  manner,  as  is  now  sufficiently  evident  by  inspection. 

85.  The  development  of  the  face  lines  of  the  intrados.  —  1st 
Method.  Parallels  to  AC  through  the  ^oints  of  division  1,  2, 
3,  etc.,  on  ABX,  are  the  developments  of  the  elements  whose 
vertical  projections  are  the  vertical  projections  1',  2',  3',  etc., 
of  the  same  points  of  division.  The  horizontal  projections  of 
the  extremities  of  these  elements  are  found  by  projecting  1',  2', 
....  6',  7',  etc.,  upon  both  face  lines,  AB  and  CD,  as  at  6 
and  7  on  AB,  and  7  on  CD.  Thence,  in  being  developed, 
they  pass  in  planes  6  —  zx ;  7  —  z2,  etc.,  perpendicular  to  the 
axis  EF,  to  zv  z2,  etc.,  of  ABM  and  from  7,  etc.,  on  CD,  to  vh, 
etc.,  on  CDi,  upon  the  developed  elements,  7  vn,  etc. 

86.  Construction  of  the  developed  face-line  as  a  sinusoid. 
(Theor.  II.)  If  the  equal  divisions  of  A'E'B'  be  projected  upon 
AB,  as  at  6,  7,  etc.,  these  points  will  divide  AB  in  a  certain 
manner.  If  the  latter  points  be  thence  projected  upon  GiBx, 
projection  of  AB  upon  G^,  the  lines  AB  and  G^  will  be 
divided  similarly  ;  hence,  if  B^m^  be  an  arc  similar  to  A'E'B', 
and  on  the  chord  B^,  it  will  be  divided  equally  by  the  paral- 
lels to  GGd  through  the  points  B  ....  6,  7,  etc.,  on  AB. 
Hence,  we  may  regard  the  arc  B^Gi,  and  the  straight  line 
B:A,  as  respectively  the  circular  projection  and  the  develop- 

5 


66  STEEEOTOMY. 

ment  of  a  helix,  points  of  whose  sinusoid  projection,  BjA,  are 
found  as  the  other  like  curves,  KPQ,  have  been,  from  like 
data  ;  viz.,  at  the  intersections  of  parallels  to  GGX,  from  mx  .  .  . 
m6,  mT,  etc.  with  parallels  tb  G^  through  like  points,  1,  2,  3, 
....  6,  7,  etc.,  on  the  straight  line  BXA. 

87.  Useful  limits  of  the  arc  of  the  intrados.  —  These  are  evi- 
dent from  the  development  while  considering  the  helical  joints. 
Were  the  arch  extended  to  a3a4  —  A0A4,  so  as  to  become  full 
centred  (27),  CDi  would  be  extended  each  way,  as  shown  at 
DiD2,  where  Did2  =  A'A0,  and  D2d2  =  a3v  ;  and  the  angle  at 
D2  is  90°.  Hence,  we  see  that  parallel  coursing  joints,  quite 
nearly  perpendicular  to  the  face  line  between  C  and  Du  as  is 
desirable,  —  become  more  and  more  oblique  to  it  as  we  approach 
D2  on  DiD2.  Moreover,  as  can  readily  be  imagined  with  a 
given  case  in  view,  the  greater  the  obliquity,  that  is  the  more 
acute  the  angle  CAB,  the  less  should  be  the  segment  taken  to 
form  the  intrados. 

88.  The  extrados.  —  While  the  outer  surface  of  the  actual 
arch  would  be  left  rough,  yet  it  is  convenient  to  represent  an 
ideal  extrados,  or  extrados  of  construction,  to  aid  in  forming 
the  voussoirs. 

Such  an  ideal  extrados  is  a  cylindrical  surface  having  the 
same  axis  as  the  intrados,  and  terminated  by  the  same  vertical 
planes  of  the  faces.  Its  projections  are,  therefore,  AiB2C1D2, 
and  the  arc  A2F'B2  concentric  with  A'E'B'.  Then  A2C2  and 
B2D2  are  the  outer  springing  lines. 

The  intrados  and  extrados  being  thus  concentric  cylinders, 
the  point,  as  qB'2,  in  which  the  generatrix,  Qq  —  B'B2,  per- 
pendicular to  the  axis  EF  —  O"  (79),  of  any  of  the  helicoids 
intersects  the  extrados,  will  generate  helices  upon  the  extra- 
dos, whose  developments  and  horizontal  projections  will  be 
found  in  the  same  manner  as  has  now  been  explained  for  the 
intrados. 

89.  Development  of  the  extrados.  —  As  the  helicoidal  sur- 
faces of  the  voussoirs  are  right  helicoids  (79),  their  elements 
are  perpendicular  to  the  axis  EF  —  O"  ;  hence  Kk  and  Qq, 
perpendicular  to  EF,  are  such  elements,  and  the  outer  helix 
(helix  on  the  extrados)  corresponding  to  the  inner  helix 
(helix  on  the  intrados)  KPQ  will  extend  from  k  to  q.  Hence, 
making  w9a2  =  A2FB2=  20'. 94  (77,  3°)  a2cu  parallel  to  B2D2 
is  the  developed  position  of  A2c  ;  and,  carrying  k  across  to  this 


STONE-CUTTING.  67 

line  at  &1?  gives  qkx  as  the  development  of  the  helix  from  q 
to  Jc. 

The  other  developed  transverse  helices,  as  bax  and  dcx,  are 
parallel  to  qkx,  and  are  similarly  divided,  to  give  the  developed 
extradosal  coursing  helices  bd4,  and  the  parallels  to  it,  corre- 
sponding to  BiIV,  and  its  parallels  on  the  intrados. 

90.  Contrast  between  the  intrados  and  extrados.  —  Here  we 
meet  with  two  points  of  difference.  First  —  As  the  pitch, 
Q&3,  is  the  same  for  both  of  the  helices,  KQX  and  qk^  while 
the  latter  is  longer,  the  angles  ovqo  and  oxoq  are  greater  than 
the  corresponding  angles  04QxOi  and  QiOj04  of  the  intrados. 
Hence  qoYo  and  its  equals  at  all  the  intersections  of  helices  on 
the  extrados,  are  less  than  Q1O4O1,  and  all  like  angles  on  the 
intrados. 

Second  —  As  the  outer  helix,  corresponding  to  an  inner  one 
from  C  to  D,  connects  c  with  d,  while  the  outer  face  line  is 
C2D2,  corresponding  to  the  inner  one  CD,  the  developed  extra- 
dosal face  line,  D2c2,  and  extreme  helix,  dcu  do  not  terminate 
at  the  same  points,  as  they  do  on  the  intrados  where  the  like 
lines  are  CvnDx  and  C7D2. 

91.  An  interesting  consequence  of  tl~  Urst  of  the  preceding 
differences  is,  that  to  equalize  the  angles  qoxo,  and  Q1O4O1,  so 
that  the  helicoidal  surfaces  should  be  normal  to  each  other 
somewhere  between  the  intrados  and  the  extrados,  the  initial 
coursing  joint,  B^v,  should  be  drawn  to  a  point,  IV,  on  the 
side  towards  D1?  next  to  x,  the  foot  of  the  perpendicular  from 
1$!  to  CDi,  whenever  x  does  not  coincide  with  one  of  the  points 
of  division  of  CD^ 

92.  The  construction  of  the  developed  face  lines.  —  This  may 
be  either  as  at  V",  etc.,  by  projection  from  1",  etc.,  to  1,  etc., 
on  CD,  and  transference  thence,  by  perpendiculars,  to  BD,  to 
elements  through  w2,  etc. ;  or  as  at  B2a2  by  the  sinusoid  projec- 
tion of  a  helix,  from  B2w5w9,  and  bau  the  latter  substituted, 
without  affecting  the  result,  for  a  straight  line,  B2a2,  as  its  cir- 
cular projection  and  development.  Either  method  is  obvious 
on  inspection,  in  connection  with  the  like  constructions  of  the 
developed  face  lines  of  the  intrados. 

The  construction  of  the  horizontal  projections  JcPq,  etc.,  and 
rSt,  etc.,  of  extradosal  helices,  is  now  obvious  by  inspection. 
Thus  m2  is  the  intersection  of  the  projecting  line  m'm2  with 
m'3m2  perpendicular  to  EF. 


68  STEREOTOMY. 

93.  Convenient  checks  upon  the  accuracy  of  the  horizontal 
projections  and  developments  of  the  helices  and  face  lines  occur 
as  follows:  First  —  The  necessary  intersection  of  correspond- 
ing inner  and  outer  helices  on  EF,  in  horizontal  projection,  as 
at  L,  S,  P,  and  N,  because  the  elements  of  the  right  helicoids 
containing  such  helices  are  vertical  at  these  points.  Second  — 
The  developed  positions  as  Lx  and  lly  or  Nx  and  nx  of  such 
points  are  in  the  same  line,  LjL^,  or  N^Nn^  perpendicular  to 
EF,  and  on  E^  and  exfx,  the  developments  respectively  of 
EF  —  E'  and  EF  —  F.  Third  — As  helices  from  C  to  D  and 
from  c  to  d  necessarily  cross  EF  at  F,  the  middle  point  of  CD 
(as  in  Fig.  50  the  helix  crosses  0il2'  at  its  middle  point,  6'), 
their  developments,  CD^  and  dcx,  cross ;  the  first,  CviiDx  and 
EiFx  at  Fx,  the  middle  point  of  CvuDi ;  and  the  other,  dc±  and 
^i/i  at/i,  the  middle  point  of  dcv 

94.  Completion  of  the  abutments.  —  These  being  alike,  the 
construction  is  shown  only  on  one.  The  back  of  the  abutment 
is  properly  stepped  by  vertical  planes,  parallel  and  perpen- 
dicular to  that  of  the  face  (which  represents  the  direction  of 
the  thrust  (62)  of  the  arch),  as  shown  at  oZu  and  mZx.  Then, 
having  regard  to  symmetry,  and  to  the  protection  of  the  cor- 
ners of  the  abutment,  make  mL  =  oZl5  and  draw  BZ  ;  and,  for 
the  opposite  end  of  the  abutment,  make  the  angle  D2D2=  ZBB2, 
and  Dz  =  BZ.  These  last  details  being,  however,  unessential, 
may  be  varied  at  pleasure. 

95.  The  face  joints.  —  These  are  the  intersections  of  the 
planes  of  the  faces  with  the  coursing  helicoids. 

Now,  referring  to  Fig.  50,  as  all  the  elements  intersect  the 
axis,  and  are  perpendicular  to  it,  any  plane  containing  the  axis 
or  a  perpendicular  to  it  at  any  point  of  a  helix,  will  contain  an 
element  of  the  helicoid. 

Comparing  with  Fig.  49,  the  plane,  AB,  of  the  face  for  ex- 
ample, does  not  contain  S,  P,  or  L ;  hence,  it  does  not  contain 
elements  of  the  helicoids  SRr,  K&P,  or  MraLI,  which  it  there- 
fore intersects  in  curves.  It  intersects  the  coursing  helicoid 
SRr  in  the  face  joint  R"V". 

96.  The  direct  construction  of  the  face  joints  will  then  be,  to 
project  the  points  as  R'"  (derived  from  R"",  extremity  of  a 
coursing  joint  on  the  developed  inner  face  line)  at  R' ;  and  r"', 
likewise  derived  from  rju  on  the  developed  outer  face  line,  at 
r' ;    when  RV,  though   straight,  would  very  nearly  be   the 


STONE-CUTTING.  69 

proper  face  joint.    And  so  we  might  operate  to  find  all  the  face 
joints. 

97.  Representation  by  tangents.  —  The  face  joints  are  so 
nearly  straight,  especially  in  a  segmental  arch,  that  their  tan- 
gents, at  their  inner  extremities,  may  often  be  sufficient  to 
represent  them.  Now  we  know  from  descriptive  geometry: 
first,  that  the  tangent  line  at  a  given  point,  t,  of  the  intersec- 
tion of  any  plane,  P,  with  a  surface,  S,  is  the  intersection  of 
the  plane  P  with  the  tangent  plane  to  the  surface  S,  at  the 
point  t ;  second,  that  the  tangent  plane  to  a  helicoid,  at  a  given 
point,  is  determined  by  the  element  through  that  point  and  the 
tangent  to  the  helix  through  that  point. 

98.  The  following  construction  depends  on  the  principles  just 
given.  Let  the  tangent  at  R'"R'  be  constructed.  First  —  AB 
is  the  plane  of  the  curve,  and  its  trace  on  the  plane,  EE„  of 
right  section  is  E  —  0"F'.  Second  —  The  tangent  plane  at 
R"'R',  to  the  helicoid,  RrS,  is  determined  by  the  element 
R'"L2  —  R'O",  and  the  tangent,  at  R'"R',  to  the  helix,  RS  — 
B'R'E.  Now  the  tangent  to  a  helix  at  a  given  point  lies  in 
the  plane  which  is  tangent  to  the  cylinder  containing  the  helix, 
along  the  element  containing  the  given  point,  and  it  makes  the 
same  angle  with  that  element  that  the  helix  does.  But  as  the 
latter  angle  is  constant  for  all  the  elements,  the  development 
of  the  helix,  upon  the  tangent  plane  must  coincide  with  its 
tangent  in  that  plane. 

Hence  the  tangent  at  R'  is  the  vertical  projection,  and 
R""R",  coinciding  with  SjRj,  is  the  development  of  the  tangent 
line  at  R'"R' ;  and  R"W"  is  the  projection,  upon  EE^  of  the 
portion  R""R"  of  this  tangent.  Hence  make  R'W'  =  R"W", 
and  W  will  be  one  point  of  the  vertical  trace,  on  the  plane 
EEi,  of  the  tangent  plane  to  the  helicoid  RrS,  at  R'"R'.  But 
the  element,  R'"L2 — -R'O",  is  parallel  to  the  vertical  plane 
EEi,  hence  WU',  parallel  to  R'O"  is  the  vertical  trace  of  the 
auxiliary  tangent  plane.  This  meets  E  —  0"F',  the  like  trace 
of  the  plane  of  the  curve,  at  U',  which  is  therefore  one  point  of 
the  required  tangent  line.  The  given  point  of  contact  R'"R'  is 
another;  hence  ER" — U'R'  is  the  tangent,  at  R"'R',  to  the 
face  joint  through  that  point. 

99.  Focus  of  like  Tangents  to  Face  Joints.  —  Draw  R'X, 
perpendicular  to  the  horizontal  0"X  ;  and  0"Uj  perpendicular 
to  U'W  and  hence  to  0"R'.     The  angles  at  U'  and  R'  in  the 


70  STEREOTOMY. 

triangles  O^UjU'  and  R'XO"  thus  formed  are  therefore  equal, 
and  those  at  Ux  and  X  are  equal,  being  right  angles.  Hence 
these  triangles  are  similar ;  whence  we  have  :  — 

0"U'  cos.  U'CUi  =  O"^  (1) 

Also,  0"U1  =  R'W  (2) 

Remembering  that  R'W'=R"W";  and  calling  the  pitch, 
=  3TV5,  of  the  entire  helix,  3RST,  =  j^,  we  have 

9-n-f>"R' 

R'W  =      h      .  W"W".  (3) 

From  the  triangle  R"'EE2 ;  R'"E2  =  EE2  tang.  R'"EE2     (4) 

But,  EE2  =  0"X.  (5) 

andO"X=0"R'cos.R'0"X;  or,0"X=0"R'cos.  U'O"^  (6) 

Now  cancelling  all  the  terms  which  are  common  to  the  two 
columns  of  left  hand  and  right  hand  members  in  these  six 
equations,  and  multiplying  together  the  remaining  terms  in 
like  columns,  we  have  : 


0"U'=27T.O"R'.  tan.  E'"EE2  ,„. 
h (7> 

Now  in  this  equation,  every  term  in  the  right  hand  member  is 
constant,  hence  0"U'  is  constant. 

We  thus  find  the  interesting  property  that  the  tangents  to 
all  the  face-joints  at  their  inner  extremities,  meet  at  a  common 
point  on  the  vertical  line  through  the  centre  E,Ou  of  the  face 
of  the  arch. 

The  point  U'  is  called  the  focus,  and  the  distance  0"U',  the 
eccentricity  of  the  arch  ;  or,  more  strictly,  of  the  intrados  ; 
for  it  is  evident  that  a  similar  construction  exists  for  the  extra- 
dos,  or  for  all  the  points  in  any  cylinder,  concentric  with  the 
intrados. 

100.  Condensed  construction  of  Foci.  —  U7  being  indepen- 
dent of  the  position  of  R'  on  A'E'B',  take  A0a3,  the  point  in 
the  horizontal  plane  of  the  axis,  and  draw  through  it  a  devel- 
oped helix  as  a^u  meeting  the  plane  of  right  section  FFX  (cor- 
responding to  EEj)  at  w,  then,  projecting  a3  on  FF2  at  o",  gives 
o"u  =  0"V. 

Likewise,  for  the  extrados,  take  A4<x4,  corresponding  to  A0a3 
in  the  intrados,  and  draw  a4v  parallel  to  the  developed  extra- 
dosal  helices,  6wi,  etc.,  and  project  a4A4  on  FF:  at  02,  then  02v 


STONE-CUTTING.  7 1 

will  be  the  eccentricity  to  lay  off  from  O",  giving  U"  as  the 
focus  for  the  extrados  ;  so  that,  for  example,  r'U"  is  the 
vertical  projection  of  the  tangent  at  r'"r'  to  the  face-joint 
R/V"  —  RV. 

101.  Foci  for  any  Cylinder  concentric  with  the  Intrados.  — 
If,  in  a  given  arch,  we  take  such  a  new  cylinder,  only  the 
terms  0"U'  and  0"R/  will  change  ;  hence,  as  is  readily  obvi- 
ous, calling  R0,  Ro>  etc.,  the  successive  values  of  the  radius 
0"R',  and  calling  the  consequent  positions  of  the  focus,  U',  U", 
etc.  we  have, 

K  _  TO2 -ctc 

0"U'  ~  0"U"  ~~ 
Then  putting  ~^.f  =  c,  we  find  0"U"  =  (-^ 

whence  0"U"  is  found  simply  as  a  third  proportional.  Thus 
having  drawn  U'A0,  and  A0T,  perpendicular  to  it,  gives  0"T 
(on  0"F  produced)  =  c  ;  since  R*  =  (O"A0)2  =  0"U"  X  0"T. 
Then  c  being  constant,  draw  for  example  TA4,  and  A4U"  per- 
pendicular to  it,  and  U"  will  be  the  foeir  of  A4F'B2,  which  is 
chosen  for  illustration,  to  save  additional  lines.     For, 

(0"A4)2=0"U"xO"T 

(0"U")  =   off-  =  ~-  as  above- 

In  this  simple  way  we  finally  find  the  focus  for  any  cylinder, 
and  thence  as  many  tangents  as  we  please  to  each  face  joint  in 
making  an  exact  working  drawing  on  a  large  scale.  In  the 
figure,  then,  the  face  joint  RV  will  be  a  curve,  tangent  to  U'R' 
at  R'  and  to  U'V  at  r'. 

102.  Curves  of  the  Face-joints.  —  To  get  an  idea  of  these, 
see  Fig.  50  again,  where  PQP',  PiQiPl  (PiQi  not  shown)  and 
P2Q2P2  are  three  parallel  planes  cutting  the  right  helicoid 
shown  by  the  shaded  area  of  the  figure. 

Auxiliary  horizontal  planes  will  intersect  both  the  helicoid 
and  any  one  of  the  given  planes  in  straight  lines,  whose  inter- 
section, in  each  of  the  auxiliary  planes,  will  be  a  point  of  the 
required  curve. 

Thus,  taking  the  plane  PQP',  the  horizontal  plane  O'Q  cuts 
from  the  helicoid  the  element  o0 — 0',  and  from  the  plane, 
the  line  PQ.      These  being   parallel,  meet  only  at   infinity ; 


72  STEREOTOMY. 

hence  PQ  is  an  asymptote  to  the  curve.  The  plane  b'V  cuts 
from  the  helicoid  the  element  ol  —  b'V,  and  from  the  plane, 
the  perpendicular  to  V  at  c',  which  meets  the  element  at  c'e. 
Likewise,  project  d'  at  d,  etc.  Then  at  o'  the  intermediate  plane 
o'a'  cuts  from  the  helicoid  the  element  oa  —  o'a',  and  from  the 
plane,  the  line  oQ  —  o'  which  meet  at  oo',  showing  that  the 
horizontal  projection  of  the  curve  passes  through  the  centre  of 
the  circle  0-3-6-12.  Next,  the  plane  6'q'  cuts  the  helicoid 
and  the  given  plane  in  parallels  at  6'  and  q'  respectively,  which 
only  meet  i '  infinity,  hence  qr  is  another  asymptote.  The 
branch,  cdoq,  meets  this  asymptote  at  infinity  towards  q  ;  while 
the  new  branch  projected  from  s' ,  t',uf .  .  .  .  e',  f  at  s,t,u  .... 
e,/ meets  the  same  asymptote  at  infinity  towards  r,  and  the 
asymptote /&,  in  the  plane,  12'/,  at  infinity  towards/. 

Thus  we  see  that  the  entire  intersection  consists  of  two 
branches  ;  and  that  when  the  given  plane,  PQP',  as  here  placed, 
cuts  the  helix  twice  on  the  same  side  of  the  axis,  as  at  h'  and  v\ 
the  curve  is  shaped  as  at  s  t  e,  and  does  not  pass  through  o  ;  but 
that  when  it  thus  cuts  the  helix  but  once,  as  at  the  point  a  little 
below  c',  the  curve  does  pass  through  o,  in  the  horizontal  pro- 
jection. 

103.  Applying  this  construction  to  the  arch  ;  AB  represents 
the  plane  PQP'  of  Fig.  50,  and  each  coursing  helicoid,  in  suc- 
cession, will  represent  the  helicoid  of  Fig.  50.  Taking  the 
helicoid,  RrS,  for  illustration,  the  intersection  of  AB  with  suc- 
cessive elements  parallel  to  Rr  would,  when  projected  upon  the 
vertical  projections,  BgB'  j  1"1',  etc.,  of  these  elements,  be  the 
vertical  projections  of  points  of  the  indefinite  face  joint  R'r' ; 
and  a  parallel  to  EF,  at  s2,  the  intersection  of  the  plane  AB 
with  the  plane  S«i,  would  be  an  asymptote  to  this  curve  which 
would  resemble  cdoq  in  Fig.  50. 

104.  Other  projections.  — In  finished  drawings  for  exhibition 
or  other  purposes,  elevations  on  planes  parallel  to  the  face,  or 
to  the  axis,  may  be  desired.  These  are  easily  made,  as  shown 
in  Fig.  51,  a  fragment  of  the  vertical  section  through  the  axis. 
A"E"B"  is  the  right  section  of  the  intrados,  and  the  parallels 
to  A"B"  through  its  points  of  equal  division  1,  2,  3,  etc.,  are  the 
vertical  projections  of  the  elements  at  1',  2',  3',  etc.,  on  the  new 
plane. 

The  portion,  RS,  of  the  inner  helix,  RST,  is  then  projected  ; 
R,  at  Rj ;  R2,  at  K2 ;  S,  at  Si,   etc.     Other  helices  could  be 


STONE-CUTTING.  73 

similarly  projected.     The  projections  of  the  half  face  lines,  EB 
and  FD,  would  be  elliptical  arcs.1 

II.  TJie  directing  Instruments.  —  These  are  — 

No.  1.  The  straight  edge,  applicable  to  any  ruled  surface, 
in  the  direction  of  its  straight  elements. 

No.  2.  The  mason's  square,  applied  wherever  two  lines,  or 
two  surfaces  are  to  be  perpendicular  to  each  other. 

No.  3.  The  pattern,  MmZjoO,  of  the  bed  of  a  top  stone  of 
the  abutment ;  also  called  an  impost  stone,  or  springer. 

No.  4.  The  modification  of  No.  3,  applicable  to  the  bed, 
DdZiZ,  of  the  springer  at  the  obtuse  corner,  D,  of  the  abut- 
ment. 

No.  5.  The  second  modification  of  No.  3,  for  marking  the 
bed,  MmZB,  of  the  springer  at  the  acute  corner,  B,  of  the  abut- 
ment. 

No.  6.  The  bevel,  A'2,A'T,  for  marking  the  joints  li,  K&, 
etc.,  of  the  skew-back  (19)  in  the  plane,  TJfO"A'v 

No.  7.  The  internal  impost  arch  square,  D"B'l',  for  marking 
right  sections  of  the  intrados  of  the  springers,  in  their  proper 
positions  relative  to  the  face  of  the  abutment. 

No.  8.  The  external  impost  arch  square,  18A^',  which  de- 
termines right  sections  of  the  extrados,  in  case  they  are  wrought 
of  a  cylindrical  form,  in  their  true  relation  to  the  level  tops,  as 
jYi,  of  the  springers. 

No.  9.  The  flexible  pattern,  OiC^Qx,  of  the  intrados,  II2J, 
of  a  springer. 

No.  10.  The  corresponding  flexible  pattern,  i'iYi#i,  of  the 
extrados,  ii2j,  of  a  springer. 

Nos.  11  and  12.  The  modifications  of  No.  10,  which,  put 
together,  equal  No.  10,  and  which  apply  to  the  partial  extra- 
dosal  surfaces  of  the  two  end  springers  ;  which  exist  in  conse- 
quence of  the  different  points,  ax  and  a2,  at  which  the  developed 
face  line,  B2a2  and  helix  baY,  terminate. 

No.  13.  The  twisting  frame,  20,  21,  22  —  20',  21',  22'.  This 
consists  of  three  rulers,  lying  in  three  planes  of  right  section, 
and,  as  shown  by  the  drawing,  coinciding  with  three  elements, 

1  While  writing  these  pages,  an  article  on  skew  arches,  by  E.  W.  Hyde,  C.  E., 
has  appeared  in  Van  Nostrand's  Magazine,  Feb.-April,  1875 ;  which  may  be 
read  with  much  interest  by  those  who  have  acquired  a  sufficient  knowledge  of  De- 
scriptive Geometry,  of  its  applications  to  the  problem  as  exemplified  in  the  mainly 
graphical  construction  which  I  have  here  given ;   and  of  higher  mathematics. 


74  STEREOTOMY. 

20  —  20' ;  21  —  21',  and  22  —  22',  of  a  coursing  helicoid,  C0c7S2. 
Their  perpendicular  distances  apart  in  a  direction  parallel  to 
the  axis,  are  given  in  plan.  Their  angles  with  the  horizontal 
plane  are  given  in  the  elevation.  From  these  data  they  can 
be  rigidly  framed  together ;  and  can  then  be  used  to  deter- 
mine elements,  in  their  true  relative  position  upon  a  helicoidal 
side  of  a  voussoir  ;  having  first  notched  upon  their  edges  their 
intersections  with  the  helix  C  S2.  In  order  that  No.  13  may  be 
shifted  along  to  determine  successive  elements,  the  perpendicu- 
lar from  20  to  22  should  be  less  than  the  length  of  the  side  of 
the  stone  taken  in  the  direction  of  the  axis,  and  hence  called 
its  axial  length. 

No.  14.  The  soffit  frame.  —  This,  used  in  case  the  intrados 
of  a  stone  is  wrought  first,  consists  of  three  parallel  pieces,  23, 
24,  25,  framed  together  in  planes  of  right  section  and  giving 
arcs  of  right  section  of  the  intrados,  as  RgSPr'",  of  a  stone. 
Their  circular  edges  can  be  notched,  by  the  aid  of  the  draw- 
ings, so  as  to  show  their  intersections  with  the  inner  helical 
edges,  as  R3S,  of  the  voussoir. 

No.  15.  The  arch  square,  e33'e4,  used  in  giving  the  cylindri- 
cal intrados  of  a  stone  from  its  helicoidal  side  when  the  latter 
is  wrought  first. 

No.  16.  A  small,  plane  bevel,  of  the  angle  between  a  cours- 
ing joint  and  an  arc  of  right  section  ;  and  held  against  the 
curved  arm  of  No.  15,  in  the  intrados,  in  order  to  guide  No. 
15  in  a  plane  of  right  section. 

No.  17.  The  helix  templet,  54c#4c4,  whose  curved  edge,  though 
circular,  will  sensibly  coincide  with  an  inner  helical  arc,  as 
B0B3,  when  placed  against  the  helicoidal  side  of  a  stone.  Mak- 
ing the  perpendicular  from  J4  to  54<?4,  equal  to  that  from  7',  for 
example,  to  the  chord  of  the  two  divisions,  as  6'-8',  the  eleva- 
tion of  an  arc  equal  to  54<?4,  we  have  three  points,  54<#4  and  c4, 
to  determine  the  required  arc.  No.  17  must  be  applied  to  give 
a  helical  edge  as  B0B3,  before  No.  15  can  be  applied,  in  succes- 
sion to-  No.  13. 

No.  18.  A  flexible  pattern,  M!M4M3B4,  of  the  development 
of  the  intrados  of  a  stone,  gives  the  three  remaining  inner  heli- 
cal edges  of  that  intrados,  as  B3M2,  M2M,  and  MB0,  after  hav- 
ing found  one,  B0B3,  by  No.  17.  Or,  after  beginning  with  No. 
14,  it  will,  by  the  aid  of  No.  16,  give  all  these  edges. 

No.  19.  Fig.  52,  is  a  bevel  frame  for  working  the  helicoidal 


STONE-CUTTING.  75 

side  of  a  stone  from  its  cylindrical  intrados,  after  having  fin- 
ished the  latter  by  Nos.  1,  14,  and  18. 

No.  20.  The  radial  joint  bevel,  Wi<?4w2,  is  used  jointly  with 
No.  17,  as  indicated  by  the  figure,  to  locate  the  corner  edges  as 
B363  of  a  stone  upon  the  indefinite  helicoidal  side. 

No.  21.  The  flexible  pattern,  p^fs^sfsi  which  replaces  No.  18, 
for  a  voussoir  which  appears  in  the  face  of  the  arch.  There 
must  be  a  pattern  of  this  kind  for  every  voussoir  of  one  end  of 
the  arch. 

No.  22.  This  is  a  bevel  varying  for  each  stone  in  the  face  of 
the  arch,  and  giving  the  angle  between  the  tangent  to  the  face 
joint,  and  the  tangent  to  an  inner  coursing  helix,  at  their  inter- 
section on  the  face-line. 

We  find  this  bevel  for  the  point  R'"R'  as  follows  :  The  tan- 
gent, R'U7,  to  the  face-joint,  meets  the  horizontal  plane,  A'B', 
at  y'2y2.  The  tangent,  R'Y2,  to  the  coursing  helix  through 
R'"R'  pierces  the  same  plane  at  Y2Y2,  where  Y2  is  thus  found. 
RaWi,  on  the  developed  intrados,  is  the  length  of  the  vertical 
projection  of  the  development  of  the  portion,  R'"^  —  R'B', 
of  this  helix  ;  hence  making  Ww'  =  R^,  and  projecting  w' 
upon  RiWiiv  at  w,  gives  R'"w,  the  horizontal  projection  of  R'w', 
the  tangent  to  the  helix  at  R'"R'.  Then  projecting  Y2  upon 
R'"w,  gives  the  trace  Y2  of  this  tangent  upon  A'B'.  Hence 
Y2y2  is  the  like  trace  of  the  plane  of  the  required  angle,  Y2R'"t/2 
—  Y2R'i/2.  To  find  this  angle  we  have  only  to  revolve  R'"R' 
about  Y2y2  as  an  axis,  into  the  plane  A'B'  at  a  point  R6,  not 
shown  ;  when  Y2R6y2  will  be  the  required  angle. 

No.  22  is  sometimes  obtained  mechanically,  while  building 
the  arch,  as  follows,  Fig.  7.     Let  ILF  be  a  portion  of  the  upper 


4   " 

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Fig.  7. 
surface  of  the  centring,  coinciding  therefore  with  the  intrados  of 
the  arch,  and  on  which  the  inner  helices  of  both  kinds,  and  the 
face-lines,  are  carefully  chalked  by  means  of  thin  flexible  rules 
bent  over  the  centring,  and  through  points  transferred  upon 
chalked  elements  from  the  drawings. 


76  STEREOTOMY. 

Let  FL  be  the  face-line,  and  IH  an  inner  coursing  joint ; 
and  let  r,  rx,  r2  be  three  arcs  of  right  section.  Then  set  up  and 
fasten  together  three  arch  squares  (No.  15)  in  the  three  paral- 
lel planes  of  r,  rx,  and  r2 ;  and  tack  on  a  strip  oh,  whose  upper 
edge  shall  coincide  with  a  coursing  helix,  by  being  equidistant 
from  the  angles  of  the  squares,  where  it  crosses  their  radial 
arms.  Then  pass  a  plumb-line,  pb,  along  oh  till,  as  at  n,  the 
bob  rests  on  the  face-line,  when  no  will  be  a  face-joint,  and  Icn 
will  be  the  required  angle. 

No.  23  is  a  pattern  of  the  face  of  a  face-stone,  as  FS1B",  one 
end  of  which  is  in  a  face  of  the  arch.  It  would  be  found,  for 
each  stone,  on  an  elevation  parallel  to  the  face. 

III.  The  Application.  —  This  division  of  the  problem  will  be 
sufficiently  illustrated  by  the  working  of  two  of  the  principal 
stones ;  viz.,  one  of  the  intermediate  springers,  and  one  of  the 
voussoirs. 

The  springer,  IYJI2e2  —  A'l^'A^Y'lO. 

This,  being  the  most  irregular  stone,  is  further  represented 
in  oblique  projection,  Fig.  53.  As  in  all  other  cases  of  irregu- 
lar bodies,  the  stone  is  conceived  to  be  inscribed  in  a  rectangu- 
lar prism,  14,  15,  16,  17  —  10,  11,  12, 13,  from  whose  corners 
the  points  of  the  given  body  may  be  located  by  rectangular  co- 
ordinates, which  will  therefore  be  seen  in  their  true  size  in  the 
oblique  projection. 

Merely  adding  that  like  points  are  lettered  or  numbered  with 
like  letters  or  numbers,  the  figure  will  mostly  explain  itself. 
Thus,  iis,  Fig.  53,  =  iis  from  the  plan ;  i3k  =  ii2  from  the 
plan ;  and  ki2  =  26i'  from  the  elevation.  Similarly  for  all 
other  points,  horizontal  distances  are  taken  from  the  plan,  and 
vertical  ones  from  the  elevation. 

•  1°.  Bring  its  bed,  ODZyo,  Fig.  53,  to  a  plane  in  the  usual 
way,  and  mark  its  form  by  No.  3. 

2°.  On  the  edges  given  by  No.  3,  as  directrices,  work  the 
lateral  vertical  faces,  DZil,  "LyiY,  etc.,  Fig.  53,  square  with 
the  bed  by  No.  2  ;  and  thence  the  level  portion,  Yji,  of  the 
top. 

3°.  With  No.  6,  mark  the  joints,  J/  and  li,  or  test  them. 

4°.  With  No.  7,  form  channels  of  right  section  in  the  cylin- 
drical face,  II2J  ;  bring  it  to  a  cylindrical  form  by  No.  1  applied 
in  the  direction  of  the  elements,  and  mark  its  edges  by  No.  9. 


STONE   CUTTING.  7T 

5°.  With  No.  8,  No.  1,  and  No.  10,  similarly  complete  the 
cylindrical  triangle,  0$,  of  the  extrados. 

6°.  Work  the  helicoidal  cheeks,  Iii2I2  and  ijIJ,  which  re- 
ceive respectively  a  side  and  an  end  of  adjoining  voussoirs,  sim- 
ply by  dividing  the  helical  arcs,  II2  and  ii2  and  JI2  and  ji2,  each 
into  the  same  number  of  equal  parts,  and  applying  No.  1  upon 
the  corresponding  points  of  division,  as  shown  in  Fig.  53. 

A  voussoir.  —  As  we  face  the  vertical  plane,  one  of  these 
stones  is,  in  detail,  as  follows  :  Mm^Bo  is  its  front  end  or  head, 
forming  a  part  of  the  transverse  helicoid  MmJjIi.  Its  opposite 
and  equal  back  end  or  head,  is  M2m253B3.  Its  right  side  or  bed, 
MwM2m2,  is  a  part  of  the  coursing  helicoidal  surface,  beginning 
at  Mm,  Its  left  hand  bed  is  the  equal  surface  Bo^B^g.  Then 
MB0M2B3  is  its  cylindrical  intrados,  whose  development  is 
M]M4M3B4,  and  whose  vertical  projection  is  B'3'.  Its  similar 
extrados  is  mbxm^z,  whose  development  is  mm'sb5bG,  and  vertical 
projection  B2'e4. 

To  form  this  stone  from  the  rough  block,  choose  a  block  in 
which  the  finished  stone  could  be  inscribed,  then  — 

1°.  By  No.  13,  form  one  of  its  indefinite  helicoidal  beds,  as 
MwM2m2.  In  doing  this,  first  cut  three  elements  on  which  the 
edges  of  No.  13  will  apply ;  then  placing  rule  20  on  the  second 
element,  and  rule  21  on  the  third  element,  rule  22  will  be  in  a 
position  to  determine  a  fourth  element  of  the  same  helicoid. 

In  the  same  way  any  number  of  other  elements  may  be 
found,  and  the  stone  between  them  cut  away  by  No.  1. 

2°.  By  No.  17  and  20,  placed  together  as  shown  on  the  plate, 
mark  the  helical  inner  edge  MM2,  in  its  proper  position  relative 
to  the  determining  elements  marked  by  No.  13. 

3°.  By  No.  15,  having  its  straight  arm  applied  to  the  ele- 
ments given  by  No.  13,  and  its  curved  arm  guided  in  a  plane 
of  right  section  by  No.  16,  applied  on  a  small  temporary  plane 
area  cut  on  the  intrados,  cut  any  desired  number  of  arcs  of 
right  section  in  the  intrados,  MB0M2B3 ;  and  finish  the  intrados 
as  a  cylindrical  surface,  by  No.  1,  placed  in  the  direction  of 
the  elements,  as  shown  in  the  plan. 

4°.  By  No.  18,  placed  in  the  cylindrical  intrados,  first 
wrought,  and  with  one  of  its  edges  coinciding  with  MM2,  mark 
the  three  remaining  edges  of  the  intrados.  These  edges  will 
serve  as  directrices  of  the  three  remaining  helicoidal  surfaces 
of  the  stone. 


78  STEKEOTOMY. 

5°.  By  No.  15,  or,  more  exactly,  by  No.  19,  Fig.  52,  which 
consists  of  Nos.  1  and  15,  very  accurately  made  and  firmly 
braced  together,  work  the  helicoidal  bed,  BoB^A ;  and  mark 
its  corner  edges,  B353  and  B^,  by  Nos.  17  and  20  together. 

6°.  Having  now  finished  all  the  faces  of  the  stone,  except 
the  extrados,  which  is  to  be  left  unwrought,  and  the  ends,  the 
latter  can  be  wrought  by  No.  2,  the  square,  from  the  intrados ; 
one  arm  being  held  in  the  direction  of  the  elements  of  the  in- 
trados, and  the  other  applied  to  corresponding  points  of  division 
of  B3M2  and  53m2,  transferred  from  the  drawings. 

In  working  a  face-stone,  as  R3R"V',  use  No.  22  instead  of 
No.  20,  No.  21  instead  of  No.  18,  and  No.  23  in  working  the 
face. 

Systems  of  Oblique  Arch  Stone-cutting. 

105.  Adhemae,  with  great  elaborateness  of  graphical  illus- 
tration and  refinement  of  detail,  devotes  no  less  than  eighty- 
three  pages  and  seven  large  plates,  to  several  different  methods 
of  cutting  the  stones  of  an  oblique  arch,  arranged  under  two 
systems. 

1st.  The  method  by  squaring,  in  which  a  rectangular  prism 
of  stone  is  first  formed,  in  which  the  finished  voussoir  can  be 
inscribed,  and  from  whose  sides,  or  edges,  those  of  the  required 
voussoir  can  be  located  with  the  square,  by  measurement. 

2d.  The  method  by  bevels.  This  method  is  illustrated  by 
the  use  of  No.  2  (6 —  9),  15,  and  19;  a  system  which  might 
be  extended  by  framing  Nos.  13  and  14  together,  or  by  adding 
to  No.  19  a  second  arch  square,  whose  straight  arm  should  be 
to  the  left  of  EE,  giving  the  relative  position  of  an  element, 
and  a  right  section  of  the  intrados,  and  an  element  in  each 
helicoidal  bed,  all  in  one  frame,  or  compound  bevel. 

106.  Choice  of  circumscribing  block.  This  point,  which 
may  perplex  a  beginner,  working  only  by  the  first  of  the  above 
methods,  may  be  settled  in  three  ways. 

First  Method.  The  sides  of  the  circumscribing  rectangular 
prism  may  all  be  either  parallel  or  perpendicular  to  the  plane 
of  right  section.  Thus,  for  the  voussoir  R3SR6/*6,  the  length  of 
the  prism  (the  prism  not  shown,  to  avoid  confusion),  would  be 
the  perpendicular  between  the  radial  elements  at  R3  and  R6r8 ; 
while  its  width  and  thickness  would  be  shown  in  elevation  by  a 
rectangle  circumscribing  5'5i2'w',  since  the  vertical  projections 
of  r7  and  r6  are  5[  and  m'. 


STONE-CUTTING.  79 

This  method  is  the  most  elementary,  requiring,  besides  the 
square  and  straight  edge,  only  Nos.  18  and  23 ;  but  it  is  obvi- 
ously very  wasteful  of  labor  and  material. 

Second  Method.  Taking  the  same  voussoir  as  before,  pro- 
ject all  its  corners  upon  a  plane  tangent  to  the  extrados  along 
the  element  mid-way  between  r6  and  r8.  Then  the  least  rect- 
angular prism  will  be  that,  one  of  whose  sides  shall  be  in  this 
tangent  plane,  and  which  shall  include  within  it  all  the  corners 
of  the  stone.  By  this  method  there  will  be  very  little  waste 
of  material  in  any  case.  And  if  this  prism  be  first  wrought, 
the  voussoir  can  be  simply  extracted  from  it  mostly  by  squar- 
ing from  the  edges  of  the  prism,  as  before. 

If,  to  save  labor,  the  voussoir  be  wrought  directly  from  the 
rough  block,  the  latter  will  be  chosen  similar  to  the  prism  here 
described,  but  larger,  and  the  method  of  cutting  will  be  that 
above  described  in  detail. 

The  price  of  labor  and  material  will  determine  whether  it 
will  be  best  to  cut  the  finished  prism,  or  to  risk  an  occasional 
spoiled  voussoir  mis-cut  directly  from  the  rough  block,  or  from 
a  block  which  by  an  error  of  choice  may  be  too  small. 

Third  Method.  Here  the  faces  of  the  circumscribing  prism 
are  all  parallel  or  perpendicular  to  the  plane  of  the  face  of  the 
arch.  This  method  is  particularly  adapted  to  a  voussoir  one 
end  of  which  is  in  the  face  of  the  arch,  and  can  therefore  be 
immediately  marked  upon  the  face  of  the  finished  provisional 
circumscribing  prism. 

107.  Adaptation  to  a  cut  stone  spandril.  In  this  case,  the 
tops  of  the  face-stone  are  finished  with  vertical  and  horizontal 
surfaces,  as  in  PI.  IV.,  Fig.  30,  and  extending  through  the 
thickness  of  the  surmounting  parapet.  The  horizontal  sur- 
faces are  plane,  and  the  vertical  ones  are  cylindrical,  having 
the  extradosal  coursing  helices  for  their  directrices.  The  cut- 
ting of  the  face-voussoirs  thus  designed,  offers  no  special  diffi- 
culty. 

Useful  Numerical  Data. 

108.  By  some  writers  the  problem  of  the  oblique  arch  has 
been  treated  almost  wholly  by  the  equations  of  its  lines,  and 
of  their  projections;  from  which  all  its  points  could  be  located 
by  merely  plotting  to  scale  the  results  of  computations,  made 
by  substituting  numerical  data  in  the  formulas.1 

l  Bashforth,  London,  1855.     Buck,  London,  1839.     Graef,  Paris,  1853. 


80  STEREOTOMY. 

But  such  treatment  is  less  adapted  for  general  use  than  the 
graphical  one,  as  most  thoroughly  exhibited  by  Adhemar  ;  and 
which  is  all-sufficient,  mainly  on  account  of  the  simplicity  and 
similarity  of  the  principal  lines,  on  which  all  the  points  depend. 
Nevertheless,  a  few  simple  formulas,  such  as  all  can  use,  are 
useful  as  checks  upon  purely  graphical  constructions,  especially 
when  the  latter  are  of  half,  or  whole  size,  upon  a  platform,  as 
they  should  be  in  practice. 

109.  Such  formulas  are  the  following,  with  the  numerical 
data,  and  results,  used  in  the  present  example,  substituted  :  — 

The  half  span  0"'B'  =  S  =  6.5  ft. 

O'"B'O"  =  a  =  30°. 

Then  0'"0"  =  S.  tan  a  =  3.75  ft. 

and  0"B'  =  E  =  VS2  +  0'"0"2  =  7.5  ft.  (7.504  ft.) 

Again,  let  the  angle  of  skew,  E AC,  =  0 

then  EAG  =  90°  —  0, 
whence  BG  =  2S.  tan  (90°  —  0)  (1) 

and  AB  =  h/AG2+BG2=  */l32  +  (2S.  tan  (90  —  Of 

oo 

or,  AB  =  2S.  sec.  (90  —  0)  =  2S.  cosec  6  =  -~  . 
v  J  sin  0 

Also  CC,  =  A'E'B'  =  I  =  R  X  .0174533  X  120°  (2) 

=  15.708  ft.; 

where  A'E'B'  =  120°, 

and  .0174533  is  the  length  of  1°  where  R=  1°. 

Likewise,  putting  0"Ba  =  Ru 
nda2  =  AjF'B;  =  l1  =  Rlx  .0174533  X  120°,  (3) 

where  Rx  =  7.5  +  2.5  =  10,  whence  nsa2=  20.94  ft. 

Further  ;  let  C2y4,  perpendicular  to  AB,  =  L  =  14  ft. 

Then,  as  A2C2^  =  (90  —  0.) 

AC  =  BA  =  A2C2  =  L.  sec  (90  —  0)  =  L.  cosec  0  =  ^ 

And,  see  (1)  and  (2),  AB:  =  CDX  =  ^AGf+B^l         (4) 

Now  suppose  CDX  to  be  divided  into  n  equal  parts  of  which 
Diiv  are  m  parts.     Then  D,iv  =  —  CDi. 

or,  in  this  example,  see  (4),  D,iv=|.  CDX. 

Let  ABxGx  =  B1D1C  =  ft  then  ^  =  tan  (3.  Or,  as  by  (1) 
B1G1  =  BG  =  2S.  cot0 


5T- 

— 

. 

2r0 

28 

| 

- 

- 

- 

- 

— 

| 
-}c- 

Jfc^f 

n 

24 

82  STEREOTOMY. 

4°.  Avoiding  the  acute  diedral  angles,  between  the  face  and 
the  intrados,  near  the  acute  angles,  as  B,  of  the  abutments,  by- 
terminating  the  cylindrical  intrados  by  a  plane  parallel,  and 
near  to  AB  ;  through  r5  for  example,  and  then  completing  the 
intrados  between  this  plane  and  AB  by  a  conoid,  whose  direc- 
trices should  be  the  ellipse  in  the  plane  r5,  and  a  vertical  line  on 
BD,  as  at  Q,  and  equal  to  0"'E' ;  and  whose  plane  director 
should  be  horizontal. 

5°.  The  substitution  of  brick  for  stone ;  either  wholly,  or  ex- 
cept in  the  face  ring  of  stones. 

6°.  The  substitution  of  a  circular  for  an  elliptic  face.  This 
will  make  the  right  section  an  arc  of  an  ellipse  whose  longer 
axis  will  be  vertical.1 

111.  Among  much  more  important  hinds  of  modification,  are 
the  following :  — 

1°.  Convergent  arches.  —  When  an  arch  is  quite  long,  the 
central  portion  of  it  may  properly,  and  with  great  economy, 
be  built  as  a  right  arch,  either  of  brick  or  stone.  Then,  near 
the  ends  the  coursing  joints  may  gradually  bend  till  perpendic- 
ular to  the  face  line  at  their  intersections  with  it. 

2°.  Hart's  System.2  —  In  this  system,  the  transverse  joint 
surfaces  are  planes,  parallel  to  the  face  ;  while  each  coursing 
joint  surface  is  a  conoid,  generated  by  a  line  moving  parallel 
to  the  face  as  a  plane  director,  and  upon  the  axis,  and  a  helical 
coursing  joint  as  directrices. 

3°.  Cylindrical  coursing  joint  surfaces.  —  This  very  inter- 
esting modification  of  the  orthogonal  or  equilibrated  system,  is 
strongly  advocated  by  Adhemar,s  first,  as  dispensing  with  all 
cut  surfaces  except  plane  and  cylindrical  ones,  so  related  as  to 
be  the  ones  most  easily  wrought  with  economy  of  material,  la- 
bor, and  graphical  construction ;  second,  as  wholly  avoiding  those 
components  of  pressure  which  act  towards  the  faces,  and  tend 
to  produce  dislocation. 

The  transverse  joint  surfaces  in  this  system  are  planes  par- 
allel to  the  face. 

The  coursing  joint  surfaces  are  the  horizontal  cylinders,  per- 

1  Such  arches,  on  the  helicoidal  system,  are  the  subject  of  a  work  by  Praly, 
Paris,  1853. 

2  John  Hart,  London,  1848. 

3  Only  regretting  that  prescribed  limits  imperatively  forbid  adequate  illustra- 
tion of  this  system ;  the  reader  is  referred  to  the  ample  exhibition  of  it  given  by 
Adhemar,  Paris,  1861. 


STONE-CUTTING.  53 

pendicular  to  the  plane  of  the  face,  which  project  the  trajec- 
tories, or  intradosal  coursing  joints  of  the  orthogonal  arch, 
Prob.  XII.,  upon  the  plane  of  the  face. 

4°.  The  Cow's  Horn.  —  The  form  of  oblique  arch  whose  in- 
trados  is  a  portion  of  the  warped  surface  so  named,  may  be 
substituted,  in  case  of  small  arches,  as  oblique  door-ways. 
See  PI.  VII,  Fig.  54. 

The  Cow's  Horn  (Come  de  Vache)  is  a  warped  surface,  of 
the  kind  having  three  given  directrices.  In  its  usual  form, 
it  is  generated  by  a  straight  line,  AC  —  A'C,  moving  so  as 
always  to  rest  upon  two  equal  and  parallel  semicircles,  AB  — 
A'N'P/,  and  CD  —  C'N'D',  whose  diameters  are  in  the  same 
plane  ;  and  upon  a  straight  line,  00"  —  O',  perpendicular  to 
the  planes  of  these  semicircles,  at  the  point  of  symmetry  O'. 

This  surface  is  the  intrados  of  the  arch.  The  semicircles 
are  its  face  lines.  The  semicircle  on  L/M'  is  the  ideal  extrados 
of  construction.  The  face-joints,  as  a'c'  and  m'p',  are  in  pairs, 
symmetrical  with  O'M' ;  the  coursing  joints,  ab  —  a'6',  etc.,  are 
elements  of  the  warped  surface ;  and  the  available  height  of 
the  passage  is  O'N'.  This  height  diminishes  with  the  increas- 
ing obliquity  of  the  arch,  till,  when  A'  and  D'  coincide,  the 
arch  degenerates  into  two  cones,  tangent  to  each  other  on  00" 
—  O'  and  the  passage  becomes  closed.  This  is  an  objection  to 
this  form  of  arch. 

Examples.  —  1°.  Make  a  full  construction,  with  patterns,  etc.,  for  the  Cow's 
Horn. 

2°.  Do.  of  a  segmental,  helicoidal  arch,  ?e/i!-handed,  that  of  Prob.  XIII.  being 
ngfa-handed,  of  12  ft.  span,  55°  skew,  120°  arc  of  right  section  and  2  ft.  radial 
thickness,  and  12' :  6"  perpendicular  between  faces. 

3°.  Do.  of  a  full  centred,  helicoidal  arch,  with  60°  angle  or  skew. 

4°.  Construct  an  arch  on  the  cvlindric  system. 

5°.  On  Hart's  system. 

6°.  A  segmental,  helicoidal,  left-handed  arch  of  14  ft.  span ;  perpendicular 
length  12  ft.,  48°  skew;  135°  arc  of  right  section;  radial  thickness  2  ft.,  2  in.,  and 
with  the  face  voussoirs  adapted  to  a  cut  stone  spandril  wall.  Add  several  oblique 
and  isometric  projections  of  springers  and  voussoirs. 

Wing-  Walls.         x 

112.  Prismatic  and  pyramidal  wing-walls  have  been  suf- 
ficiently illustrated  in  outline  on  PI.  I.,  Figs.  6-9.  The  vertical, 
quarter-cylindrical  •  one,  with  an  oblique  plane  top  as  in  PI.  L, 
Fig.  8,  is  described  in  my  "  Elementary  Projection  Drawing." 
The  top  of  the  straight  wall,  which  contains  the  face  of  the 


84  STEREOTOMY. 

arch,  or  passage,  and  which  is  flanked  by  the  wing-wall,  is  some- 
times horizontal :  while  either  wing-wall  is  a  segment  of  a 
cylinder,  oat  off  obliquely,  as  in  PL  L,  Fig.  9.  The  top  of  either 
wing-wall  is  then  a  portion  of  a  right  helicoid,  having  a  vertical 
axis  ;  that  is,  if  such  a  wing-wall  were  a  quarter-cylinder,  its 
coping  would  be  one  quarter  of  a  revolution  of  a  square-threaded 
screw  of  stone. 

All  the  foregoing  forms  are  in  use  in  the  wing-walls  of  small 
arches,  culverts,  and  mounded  cemetery  vaults. 

There  yet  remain,  however,  two  forms  of  wing-wall,  adapted 
to  larger  structures,  and  which  will  now  be  described. 

113.  PI.  VIII.,  Fig.  55,  is  a  sketch,  in  plan  and  elevation,  of 
the  crossing  of  two  lines  of  communication,  at  different  levels, 
and  at  an  acute  angle.  The  lower  one,  rr,  passes,  by  an  oblique 
arch,  flanked  by  wing-walls,  AC  and  BCx,  through  the  embank- 
ment, MM,  which  supports  the  upper  line.  The  axis,  parallel 
to  rn,  of  the  arch,  makes  an  angle,  Cnr,  with  the  direction, 
CCi,  of  the  embankment  through  which  the  arch  runs. 

This  understood,  it  is  determined  that  the  wing-walls  shall 
extend  a  given  distance  in  front  of  Cd,  without  being  as  far 
apart  at  A  and  B  as  they  would  be  if  wholly  curved.  They 
are,  therefore,  in  plan,  made  partly  straight  and  partly  curved, 
with  a  quite  small  radius,  as  Oo. 

The  straight  wall,  containing  one  face  of  the  arch,  lies  between 
the  vertical  planes  CO  and  Cft^  Then,  if  the  face  of  the  wall 
slopes,  that  of  the  curved  portion  of  each  wing-wall  will  be  a 
segment  of  a  cone  having  a  vertical  axis  ;  and  the  faces,  both  of 
the  arch-wall  and  of  the  straight  part  of  the  wing-wall,  will 
be  tangent  planes  to  this  cone.  All  parts  of  the  face  of  the 
wall  will  thus  have  the  same  batter. 

From  the  complete  plan,  it  will  be  seen  that  if  the  straight 
parts  of  the  two  wing-walls  are  to  be  parallel  to  the  centre  line 
of  the  arch,  and  equidistant  from  it,  the  conical  portions  must 
be  unequal  segments,  and  of  unequal  radii,  the  larger  segment, 
adjacent  to  Cx,  having  the  less  radius. 


STONE-CUTTING.  85 

Problem  XIV. 

The  compound,  or  piano-conical  wing-wall. 

I.  The  Projections.  1°.  Preliminaries. —  PL  VIII.,  Fig.  56, 
represents  the  plan  and  elevation  of  a  wing-wall,  similar  to  that 
part  of  Fig.  55  which  is  to  the  left  of  CO  ;  but  with  the  ad- 
dition of  a  pier  at  the  foot  of  the  wall,  and  a  coping  ;  also,  the 
plane  V?  instead  of  being  taken  as  in  Fig.  55,  parallel  to  CCX, 
is  parallel  to  a  plane,  as  OC,  perpendicular  to  CCV 

Let  the  vertical  line  at  O  be  the  axis  of  the  inverted  cone  of 
revolution,  a  part  of  whose  convex  surface  is  the  face  of  the 
conical  part  of  the  wall. 

The  circle  with  radius,  OA,  of  four  feet,  is  the  horizontal 
trace  of  this  cone,  and  the  arc  AG,  of  60°,  is  that  of  the  conical 
portion  of  the  wall.  The  tangents  to  AG  at  A  and  G  are  re- 
spectively the  horizontal  traces  of  the  planes  of  the  faces  of  the 
arch- wall,  and  of  the  straight  portion  of  the  wing-wall ;  which 
are  respectively  tangent  to  the  conical  part,  along  the  elements 
whose  horizontal  projections  are  VB  and  VGT.  Let  the  height 
of  the  wall  under  the  coping  be  6  ft.  4  in.  =  76  ins.  and  the 
batter,  £.  The  horizontal,  AB,  of  the  batter  is,  then,  19  ins.  ; 
and,  making  BC,  the  thickness  of  the  wall  at  top,  2  ft.,  we 
reach,  at  C,  the  back  of  the  wall. 

2°.  Outlines  of  the  elevation.  —  We  must  now  turn  to  the 
elevation,  and  make  D'C  =  6  ft.  4  ins.  ;  draw  C'B'  parallel  to 
the  ground  line  RD',  and  project  A  at  A'  and  B  at  B'.  Then 
A'B'C'D'  is  the  section  of  the  wing- wall  at  its  junction  with 
the  arch- wall,  in  the  vertical  plane  OC.  Hence  V,  the  vertical 
projection  of  the  vertex  of  the  conical  portion  of  the  face  of 
the  wall,  is  found  by  producing  the  element  BA — B7A',  parallel 
to  Vi  till  it  meets  the  axis  O'O. 

The  section  of  the  coping,  in  the  vertical  plane  OC,  is 
C'E'F'H' ;  8  inches  thick  by  2  ft.  3  inches  wide. 

Next,  suppose  the  plane  slope,  QC,  of  the  embankment, 
behind  the  wing- wall,  to  be  one  of  \\  to  1,  and  to  contain  the 
back  top  edge,  QC — C,  of  the  arch-wall.  Accordingly,  ma-ke 
D'Q  =  |  D'C,  then,  as  the  plane  of  the  slope  of  the  em- 
bankment is  perpendicular  to  the  plane  OC,  Figs.  55  and  56,  its 
horizontal  trace,  as  AB,  Fig.  55,  PQ,  Fig.  56,  is  parallel  to  COx. 
Hence  PQC,  Fig.  56,  is  the  plane  of  this  slope. 


86  STEREOTOMY. 

3°.  Conditions  for  the  design  of  the  top  of  the  wall.  —  Let 
these  be  three,  as  follows  :  — 

1st.  Its  front  and  back  edges  shall  be  plane  curves  ;  sections 
of  the  front  and  back  surfaces  of  the  wall. 

2c?.  It  shall  be  of  uniform  width,  as  seen  in  horizontal  pro- 
jection. 

3d.  Its  straight  elements  shall  be  horizontal. 

Let  the  plane  SRB',  parallel  to  PQC,  be  that  of  the  front 
top  edge,  whose  vertical  projection  is  therefore  in  RB'.  To 
find  its  horizontal  projection,  aTB,  proceed  as  in  Des.  Geom., 
Prob.  LVII.  Thus,  projecting  G  at  G',  gives  VGT— V'G'T' 
as  one  of  the  limiting  elements  of  the  conical  face  of  the  wall, 
and  T',  its  intersection  with  RB',  is  the  vertical  projection  of  a 
point  of  the  required  edge,  whose  horizontal  projection,  T,  is 
therefore  the  projection  of  T'  upon  VG  produced.  In  like 
manner,  any  other  points  can  be  found.  The  front  upper  edge, 
TB-T'B',  of  the  conical  part,  is  an  elliptical  arc,  whose  horizon- 
tal projection,  TB,  is  also  elliptical,  but  in  this  case  so  slightly 
so,  that  it  may  be  represented,  without  sensible  error,  by  a 
circular  arc  whose  centre,  k,  on  OA,  may  be  found  by  trial. 

The  front  edge,  aT,  of  the  straight  portion  of  the  wall,  is 
tangent  at  TT'  to  TB — T'B',  it  being  the  intersection  of  the 
tangent  plane,  GS,  to  the  conical  part,  along  the  element  GT, 
with  the  plane,  SRB',  of  the  top  edge.  Hence  (Des.  Geom. 
172)  S  is  a  point  of  this  tangent,  which  is  therefore  projected 
in  ST— R'T'. 

Supposing  the  wall  to  be  terminated  at  a\  at  a  height,/*/"', 
of  20  ins.  from  the  ground ;  project  a!  upon  ST  at  a,  make 
af  =  BC  =  2  ft.,  by  condition  2° ;  project/  upon  the  horizontal 
a'f  at/',  and /'C  will  be  the  vertical  projection  of  the  back  top 
edge  of  the  wall,  by  condition  1°.  By  condition  2°,  draw  CK 
with  centre  k  (in  general,  a  curve  at  a  constant  normal  distance 
from  BT),  and  limited  by  &K,  and  K/  will  be  parallel  to  aT. 

Now,  by  condition  3°,  draw  T'h'  parallel  to  the  ground  line, 
till  it  meets  f'C  at  h' ;  project  h'  at  h  upon  the  curve  KC,  and 
Th — T'h'  will  be  one  element  of  the  top  of  the  wall.  Again  : 
project  K  at  K",  draw  K'7',  project  V  (which  is  on  a'B')  at  I  on 
aT  ;  and  Kl — K"l'  will  be  another  element  of  the  top  of  the  wall ; 
which,  as  can  now  be  seen,  consists  of  three  different  warped 
surfaces,  as  follows. 

By  condition  3°,  all  have  the  plane  H  as  their  common  plane 
director.     Then,  — 


STONE-CUTTING.  87 

1S£.  K/ — K"/'  and  al-a'l',  both  being  straight,  are  directrices 
of  aflK — a'fl'K",  which  is  thus  a  hyperbolic  paraboloid  (Des. 
Geom.  278). 

2c?.  KA — K"h'  being  curved,  while  IT — I'T'  is  straight,  these 
are  the  directrices  of  a  small  conoidal  portion,  KhTl — K"h'T'V. 
(See  114,  p.  92.) 

3d.  TABC — T'h'WO  is  a  warped  surface  without  specific 
name,  of  the  kind  having  two  directrices,  TB — T'B'  and 
Ck — Oh'  and  a  plane  director,  which  is  H« 

These  surfaces  are  evidently  tangent  to  each  other  along 
KZ— K»V  and  Th—Vh!. 

4°.  The  pier.  —  This  is  rectangular  in  plan,/c,  being  one  side, 
and  perpendiculars  to  it  at  e  and/,  3  ins.,  longer  than  ef,  being 
the  adjacent  sides ;  so  that  the  coping  stone,  odg,  of  the  pier 
shall  be  a  square  of  2  ft.  6  ins.,  overhanging  on  three  sides.  The 
top  of  this  stone  is  finished  as  a  square  pyramid  8  ins.  high. 

The  pier  having  no  batter,  project  e  at  e'  ;  and  the  vertical 
at  e'  is  a  corner  of  the  pier,  and  e'a'  is  the  intersection  of  its 
back  with  the  face  of  the  wall.  Project  d  in  the  vertical  at  d', 
for  the  corner  of  the  pier  ;  and  c  in  the  vertical,  c'c",  for  the 
junction  of  the  pier  coping  with  the  front  of  the  wall  coping  ; 
both  being  vertical  surfaces.  Then  c'F',  and  the  parallel  through 
H',  are  the  upper  and  lower  front  edges  of  the  wall  coping. 

Coping  edges.  —  FLc — c"YL',  being  the  intersection  of  the 
warped  top  surface  of  the  wall,  produced,  by  a  vertical  plane  and 
cylinder,  whose  combined  horizontal  trace  is  FLc,  is  not  strictly 
a  plane  curve ;  but  it  is  so  near  to  the  plane  curve,  BTa — Wa', 
that  it  may  be  considered  as  one  without  sensible  error  ;  espe- 
cially in  practical  cases,  where  OA  would  be  several  times  4 
ft.,  and  hence  the  changes  of  form  of  all  the  curved  surfaces 
much  less  quick.  F'LV  would  be  strictly  constructed  by  pro- 
jecting points  of  FLc  upon  the  vertical  projections  of  the  same 
horizontal  elements,  Kl — K"l',  etc.,  on  which  they  were  taken. 
The  other  edges  of  the  coping  are  FLc — FV  and  CKf—E'f. 

5°.  The  wall  joints.  —  In  the  face  of  the  wall,  the  coursing 
joints  of  the  conical  part  are  horizontal  circles,  as  b'r' ;  right 
sections  of  this  portion.  Those  of  the  straight  part  are  parallel 
to  Ge.  The  heading  joints  are  elements  of  the  conic  portion, 
and  parallel  to  GT  on  the  straight  portion. 

The  bed  surfaces.  —  To  avoid  acute-angled  edges  of  stones  by 
making  adjacent  surfaces  mutually  perpendicular,  the  bed  sur- 


88  ■  STEREOTOMY. 

faces,  that  is,  those  extending  from  AG,  br,  etc.,  to  the  back 
of  the  wall,  may  be  formed  in  two  ways. 

First.  If  two  lines,  as  V'B'  and  v'j'j",  perpendicular  to  each 
other,  as  at  j'y revolve  about  the  common  vertical  axis  V — W, 
which  they  both  intersect,  V'B'  will  generate  the  conical  front 
of  the  wall;  v'j'j"  will  generate  a  conic  bed  surface,  whose 
vertex  is  "W,  and  everywhere  normal  to  the  conical  front  of 
the  wall,  and  jj'  will  describe  the  horizontal  circle,,  jp — j'p', 
intersection  of  these  two  cones,  as  desired  for  a  coursing  joint. 
With  like  results,  revolve  b'b",  and  the  other  parallels  to  v'j", 
about  the  same  axis,  V — W. 

Second.  If  VV  be  made  the  centre  of  a  series  of  spheres,  of 
radii  V'A',  Y'b',  Vj',  etc.,  each  of  them  will  be  normal  to  the 
conical  face  of  the  wall,  at  their  intersections  with  it,  since  the 
elements  of  the  wall  are  radii  of  the  spheres ;  also  they  will 
intersect  the  conic  face  in  horizontal  circles,  as  before. 

Preferring  the  former  system  as  simpler,  let  the  beds  of  the 
stones  of  the  conic  wall  be  conic  surfaces  generated  by  v'j'j", 
etc.,  and  let  the  ends  of  the  stones  be  in  vertical  planes,  VB, 
VI,  etc.,  radiating  from  the  axis  V-VV  of  the  conical  wall. 

Thus,  sqrpJJ  and  s'q'r'p'I'J'JJ'W  are  the  complete  projec- 
tions of  one  stone  of  the  conical  wall. 

The  beds  of  the  stones  in  the  straight  wall  are  planes,  tan- 
gent, as  on  r'W  and  p'J',  to  the  conical  beds  in  the  conical 
wall.    Their  heads  are  in  vertical  planes  parallel  to  VGT. 

The  coping  joints.  —  These  simply  divide  the  top  of  the 
coping  suitably,  so  as  to  break  joints  with  the  upper  stones 
of  the  wall,  and  are  in  vertical  places  through  the  horizontal 
.straight  elements  KL  —  K'L',  MN  —  M'N',  etc.,  of  the  top  of 
the  coping. 

Thus,  MN  —  M'N'P'Q',  LK  —  L'K'K"L",  etc.,  are  vertical 
joint  surfaces  of  the  coping. 

6°.  Location  and  limitation  of  the  face  joints.  —  It  is  desirable 
that  a  coursing  joint  should  coincide  with  the  to£  of  the  pier, 
but  not  always  convenient  to  make  a  suitable  common  divisor 
of  the  heights  of  the  pier  and  wall ;  hence  arises  the  problem  : 
to  arrange  the  stones  in  courses  of  diminishing  thickness  up- 
wards, and  so  that  a  coursing  joint  shall  coincide  with  the  top 
of  the  pier.  Now  parallel  lines  will  be  divided  proportionally 
by  lines  which  meet  at  a  point,  P  ;  hence,  if  the  latter  lines 
divide  a  given  line,  A,  equally,  they  will  divide  a  line,  B,  not 
parallel  to  A,  unequally. 


l'l.  IV 


5 

J^^ 

3 
^1 

UB 

SoitZ. 

SuJO. 

SoJL 

s.     ir« 

a.. 

1>, 

STONE-CUTTING.  89 

Hence,  choosing  in  this  example  five  courses  of  stones,  as- 
sume any  point  as  a'  on  the  level  a'b'  of  the  pier,  drop  a  per- 
pendicular B'B"  from  B',  and  draw  a'B'  and  a'B".  This  done, 
adjust  any  convenient  edge  scale  of  equal  parts,  by  trial,  till 
the  0  of  the  scale  being  placed  on  a'B",  the  point  1  of  the  scale 
shall  fall  on  a'b',  and  5  of  the  scale  on  a'B'.  Holding  the  scale 
in  this  position,  prick  off  the  other  points  of  division  from  0  to 
5  upon  the  paper,  and  draw  lines  from  a'  through  the  points  so 
found,  to  meet  B'B",  as  at  y,  etc.  Then,  through  y,  etc., 
draw  the  horizontal  joints. 

To  avoid  acute  intersections  of  the  coursing  joints,  q'p',  etc., 
with  the  top  edge,  a'B',  of  the  wall,  the  former  must,  as  shown, 
terminate  on  convenient  heading  joints,  as  b'a'  on  u'x',  q'p'  on 
G'T',  etc. 

II.  The  Directing  Instruments.  These,  besides  Nos.  1  and 
2,  as  in  all  other  problems,  are  as  follows  :  — 

No.  3,  a  bevel,  set  to  the  diedral  angle  made  by  the  slant 
height  of  the  pyramid,  odg,  with  the  vertical  surfaces  of  the 
pier  coping.  This  angle  is  here  shown  by  revolving  the  verti- 
cal plane  of  this  bevel  till  horizontal,  as  at  oufYa. 

No.  4  is  a  pattern  of  each  face,  as  odg,  of  the  pyramid,  found 
by  revolving  it  about  dg  till  horizontal,  giving  o'"dg,  where 

o'"f1  =  oy1. 

No.  5  is  the  twisting  frame  for  working  the  top  of  the  coping 
of  the  straight  wall.  It  consists  of  two  rulers  framed  together 
so  as  to  be  in  parallel  planes,  and  so  as  to  give  the  relative 
position  of  the  edges,  as  Zc  —  Z'c'  and  Yf —  Y'f",  of  the  front 
and  back  of  this  coping.  These  rulers  are  shown  in  their  real 
relative  position,  in  the  separate  figure  (No.  5),  by  making  c3 
and  c2  horizontal  and  equal  respectively  to  cZ  and/Y  on  the 
plan  ;  and  then  3z  and  2Y,  perpendicular  to  <?3  and  <?2,  and 
respectively  equal  to  3Z'  and  2Y'  on  the  elevation.  The  per- 
pendicular distance  of  the  rule  cYY"  behind  czz"  is  af;  so  that 
the  former  applies  to  the  edge  Yf  —  Y'f"  of  the  stone,  while 
the  latter  applies  to  Zc  —  Z'c'. 

A  similar  frame  can  be  made  for  the  next  stone,  YL,  of  the 
coping. 

No.  6  is  an  elliptical  templet  fitted  to  the  front  top  edge  of 
the  conic  wall.  It  is  found  simply  by  revolving  the  elliptic 
arc,  BT  —  B'T',  around  its  transverse  axis,  BO  —  B'a',  till 


90  STEREOTOMY. 

parallel  to  the  plane  V,  when  ordinates,  as  N"MK  to  this  axis, 
will  appear  as  at  w/"N2,  perpendicular  to  a'W,  and  in  their 
real  size,  since  N"NX  is  horizontal,  and  therefore  seen  in  its 
real  size. 

No.  7,  not  shown,  is  similar  to  No.  6,  and  similarly  found, 
and  shows  the  real  form  of  the  curves,  as  FLa  —  F'Li,  of 
the  front  of  the  coping,  assumed  as  plane  curves  because  so 
near  to  B'T'  (4°).  Hence,  the  ordinates  for  finding' No.  7  will 
be  laid  off  on  perpendiculars  to  FV,  and  equal  to  their  hori- 
zontal projections,  which  are  perpendiculars  from  FL:  to  BO. 

No.  8,  the  dip  bevel,  shows  the  inclination  of  the  elements, 
b'b",  etc.,  of  the  conical  beds  of  the  stones  to  the  horizontal 
plane. 

No.  9,  the  coursing  joint  templet,  is  one  of  a  set,  one  for 
each  coursing  joint  being  evidently  required  by  the  increasing 
radii  of  the  latter.  Its  acting  edge,  ism,  is  cut  to  the  circular 
arc  of  a  coursing  joint. 

No.  10  is  flexible,  and  gives  the  relative  position  of  a  radial 
joint,  as  j"j',  and  a  coursing  joint,  as  j'p'  ;  and  hence,  when 
laid  flat,  gives  the  angle  between  an  element  and  an  arc  of  the 
developed  right  section  of  a  bed  cone. 

There  is  thus  one  for  each  bed,  applicable  to  the  two  conic 
surfaces,  top  of  one  stone  and  bottom  of  the  next  above  it, 
which  unite  on  that  bed. 

Thus,  in  the  one  shown,  the  curved  edge  of  radius  v'j'  is  an 
arc  of  the  development  of  the  horizontal  circular  section,  jp  — 
j'p',  of  the  bed  cone  whose  vertex  is  v',  while  its  straight 
edge  coincides  with  0/  —  v'j',  produced. 

No.  11  shows  in  like  manner  the  relative  position  of  a  head- 
ing and  a  coursing  face  joint,  as  s'q'  and  p'qf  ;  hence,  when  laid 
flat,  its  curved  edge,  in  the  example  shown,  is  drawn  with  the 
radius  V'j',  slant  height  of  the  face  cone  to  p'j',  as  a  base  or 
right  section,  while  its  straight  arm  coincides  with  the  ele- 
ment V'j'. 

No.  12  is  a  twisting  frame  for  the  curved  portion  of  the 
coping.  It  consists  of  three  or  more  rulers,  framed  together, 
each  in  a  vertical  plane,  and  so  that  their  acting  edges  have 
the  relative  position  of  three  or  more  elements,  as  at  X7  — 7' ; 
L^  —  LiKi,  and  LK  —  L/K',  of  the  top  of  the  coping. 

No.  13,  used  in  connection  with  No.  8,  as  indicated,  gives 
the  vertical  joints,  as  L'L",  of  the  front  of  the  coping. 


STONE-CUTTING.  91 

No.  14,  a  bevel  between  the  bed  and  back  of  a  stone,  serves 
to  keep  these  surfaces  in  their  proper  relative  position,  so  far 
as  the  back,  being  exposed,  is  wrought. 

III.  The  Application.  This,  so  far  as  not  already  sufficiently 
understood  from  the  description  of  the  directing  instruments, 
can  be  sufficiently  illustrated  in  the  working  of  a  stone  from 
the  body  of  the  conical  wall,  and  of  the  top  of  the  wall  as  a 
whole. 

Take  for  illustration  the  stone  rspqIJ  —  r's'p'q'XJ'J' ;  but  as 
it  is  obscurely  represented  in  the  general  projections,  see  Fig. 
57,  an  oblique  projection  of  it,  which  explains  itself  by  letter- 
ing like  points  with  the  same  letters  as  in  Fig.  56. 

As  in  all  other  cases,  points,  as  ii,  not  in  the  edges  or  faces 
of  the  auxiliary  circumscribing  rectangular  prism  rxurz  —  r2TPI", 
are  found  by  one  or  more  ordinates,  perpendicular  to  those 
faces,  and  to  each  other ;  and  hence  seen  in  their  real  size  in 
Fig.  57,  where  all  parallels  to  rtrs,  r±p",  and  rxu,  are  shown  in 
their  real  size. 

Thus,  in  the  two  figures  rxi2  =  rxi2 ;  i2ix  =  i&i  (plan)  and  {$ 
=  «!*'  (elevation). 

Having  then  chosen  a  rough  block  capable  of  containing 
the  finished  stone,  proceed  as  follows  :  — 

1°.  Work  a  portion  as  mmxnnx  of  its  upper  surface  to  a  plane, 
and  on  this  temporary  plane  portion  (sometimes  called  a  sur- 
face of  operation)  mark  the  circular  front  edge,  pq,  by  No.  9. 

2°.  By  No.  8,  having  its  straight  arm  in  the  temporary 
plane,  mpnq,  and  radial  to  the  curve,  as  shown,  form  any  num- 
ber of  elements  of  the  conical  top,  pJIq,  of  the  stone  ;  and 
gauge  the  thicknesses,  as  ql,  etc.,  on  these  elements,  Fig.  56, 
which  will  give  the  back  edge,  IJ,  of  the  stone. 

3°.  By  No.  10,  placed  as  shown,  mark  the  radial  edges,  ql 
and  pJ,  of  the  ends  of  the  stone. 

4°.  Work  the  conical  front,  square  with  the  top,  by  No.  2, 
the  square,  as  shown,  placing  its  arms  to  coincide  with  elements 
of  both  surfaces,  and  mark  the  vertical  joints,  sq  and  rp,  by 
No.  11,  placed  as  shown. 

5°.  The  bottom  edge,  rs,  is  equidistant  from  pq,  measured 
on  elements,  or  it  may  be  marked  by  one  of  the  set  No.  9,  or 
by  one  of  No.  11,  or  yet  again  by  No.  15  (not  shown,  but 
readily  found)  the  development  of  the  entire  front  face,  rspq — 


92  STEREOTOMY. 

r's'p'q'  considered  as  on  the  convex  surface  of  the  cone,  VV, 
of  the  face  of  the  conical  wall. 

6°.  The  ends  are  wrought  simply  by  the  straight-edge,  from 
their  edges,  already  found,  as  directrices.  The  bottom  bed  is 
square  with  the  front,  and  also  tested  by  the  suitable  form  of 
the  set,  No.  10. 

7°.  The  top  of  the  wall.  —  Having  finished  the  body  of  the 
wall,  except  the  top,  its  front  edge,  BTa  —  B'a',  can  be  imme- 
diately located  by  measuring  on  the  heading  joints,  from  any 
one  coursing  joint,  as  a'bf,  or  from  a  firm  platform  built  at  any 
suitable  level,  as  a  horizontal  reference  plane.  Having  thus 
the  edge,  Wa',  as  a  directrix,  any  number  of  level  chisel  lines 
can  be  cut  in  the  top  of  the  wall,  by  the  aid  of  a  spirit-level, 
till  the  top  of  the  wall  is  finished. 

Then,  having  wrought  the  top  of  the  coping,  by  the  forms  of 
Nos.  5  and  12,  suited  to  each  stone,  gauge  the  coping  every- 
where of  a  uniform  vertical  thickness,  which  may  be  done  by 
forms  of  No.  13,  suited  to  different  positions  along  FLc.  The 
under  surface  may  then  be  wrought,  either  by  suitable  forms  of 
Nos.  5  and  12,  or  even  by  No.  1  only. 

When  material  and  labor  are  cheap  enough,  the  somewhat 
difficult  stones  of  the  coping  of  the  conical  wall  might  be 
wrought  by  the  method  of  squaring  (105)  from  a  completely 
finished  circumscribing  rectangular  prism. 

Examples.  —  1.  Construct  a  front  elevation  of  the  wing- wall. 

2.  Construct  an  isometric  drawing  of  a  stone  from  the  body  of  the  conic  wall. 

3.  Construct  an  isometric  view,  and  an  oblique  projection  of  a  coping  stone,  to 
illustrate  the  cutting  of  it  by  squaring. 

4.  Work  out  the  entire  problem  for  the  opposite  wing- wall,  CiB,  Fig.  55. 

5.  Work  out  the  problem  of  the  vertical  quarter  cylindrical  wing-wall  with  a 
right  helicoidal  top  (whose  elements  will  therefore  be  horizontal). 

The  Conoid. 

114.  The  conoid  is  a  warped  surface,  which  is  generated  by 
a  straight  line  which  moves  upon  a  straight  line  and  a  curve 
as  directrices,  and  always  parallel  to  a  given  plane,  called  its 
plane-director.        , 

In  its  simplest  form,  partly  shown  in  PI.  VIII.,  Fig.  58, 
which  is  an  oblique  projection,  the  directrices  are  a  straight 
line,  OF,  and  a  circle  as  that  on  AC,  both  perpendicular  to  the 
plane  director,  RCO,  the  plane,  FOQ,  being  also  perpendicular 
to  that  of  the  circle.     The  plane  RCO  is  H>  and  RBC  is  V- 


STONE-CUTTING.  93 

115.  Elliptic  sections.  —  Every  plane  section  parallel  to 
ABC  is  an  ellipse.  Let  atb  indicate  such  a  section.  Since 
te  =  dq,  and  TE  =  DQ,  we  have  from  this,  and  the  triangle 
OAQ,  TE  :  AQ  :  :  te  :  aq,  which  expresses  a  property  of  two 
ellipses  having  an  axis  in  each,  equal.     Hence  atb  is  an  ellipse. 

116.  Tangents.  —  If  a  line  moves  upon  three  fixed  lines,  it 
can  have  but  one  position  at  each  point  of  any  one  of  these 
lines.  But  a  hyperbolic  paraboloid  consists  of  two  sets  of  ele- 
ments, all  those  of  each  set  parallel  to  one  plane,  and  each 
element  of  each  set  intersecting  all ;  and  hence,  any  three  of 
the  other  set.  Hence,  if  we  take  any  three  lines,  as  RK,  rJc, 
and  OF,  tangent  at  points  on  the  same  element,  as  TH,  of  the 
conoid,  and  parallel  to  one  plane,  \f,  they  will  be  elements  of 
one  generation  of  a  hyperbolic  paraboloid,  whose  other  genera- 
tion is  formed  by  moving  TH,  upon  these  tangents.  TH  being 
parallel  to  H,  will  remain  so,  and  hence  RrO  is  one  of  its  posi- 
tions, and  KF  another.  That  is  KQ  =  FO.  Thus  RTKFHO 
is  a  hyperbolic  paraboloid,  tangent  to  the  conoid  along  TH, 
and  having  H  and  V  for  the  plane  directors  of  its  two  genera- 
tions. 

117.  Normal  surface.  —  But  as  we  could,  in  the  last  article, 
have  taken  any  other  tangent  at  T,  and  parallels  to  it  at  all 
points  of  TH,  there  may  be  an  indefinite  number  of  tangent  hy- 
perbolic paraboloids  along  TH.  Of  these,  one  will  contain  all 
those  tangents  which  are  perpendicular  to  TH.  Now,  let  all 
these  latter  tangents  be  revolved  90°,  about  TH  as  an  axis,  and 
they  will  all  become  normals  to  the  conoid  on  TH.  But  as  they 
do  not  thus  change  their  position  relative  to  each  other,  they  will 
still  form  a  hyperbolic  paraboloid,  which  is  thus  the  normal  sur- 
face along  a  given  element. 

118.  General  conclusion.  —  No  distinctive  property  of  the  co- 
noid, but  only  those  of  the  hyperbolic  paraboloid,  having  been 
employed  in  this  demonstration,  this  shows  that  the  result  is  gen- 
eral ;  viz.,  that  the  normal  surface  to  any  warped  surface  at  a 
given  element,  is  a  hyperbolic  paraboloid. 


94  STEREOTOMY. 


Problem  XV. 


The  eonoidal  wing-wall. 

I.  The  Projections.  —  This  novel  form  of  wing-wall  ia 
founded  in  the  idea,  that,  at  the  foot,  FC,  of  the  wall,  where 
the  pressure  of  earth  from  behind  is  slight,  no  batter  would  be 
necessary  to  give  increased  stability  by  increased  thickness  at 
the  base ;  while,  on  the  other  hand,  at  AB  —  A'B',  where  the 
height  is  greatest,  the  need  of  a  slope  or  batter  to  the  front 
would  also  be  greatest. 

1°.  To  fulfill  these  conditions,  let  the  face  of  the  wall  be  a 
quarter  of  a  right  conoid  having  the  quadrant  EF  —  E'F'  for 
its  curved  direction  ;  the  perpendicular,  OF  —  O',  to  the  plane 
V,  for  its  straight  directrix  (O'  being  the  intersection  of  FO 
with  B'E',  produced,  where  B'E'  has  a  slope  of  3  to  10),  and  the 
plane  V  f°r  its  plane  director. 

Hence,  divide  EAF,  a  quadrant  of  9  ft.  radius,  into  any 
convenient  number  of  parts,  and  OE,  Oig,  OJi,  etc.,  drawn 
through  the  points  of  division,  and  parallel  to  the  ground-line 
D'F',  will  be  the  horizontal  projection  of  elements  of  the  eo- 
noidal face  of  the  wall. 

2°.  The  vertical  projections  of  these  elements  will  be  found 
by  projecting  E  at  E',  g  at  g',  etc.,  and  drawing  E'O'  with  a 
batter,  Wb',  of  3  to  10,  then  g'O',  etc.  But,  in  this  figure,  O' 
is  thus  made  inaccessible  ;  hence  proceed  as  follows :  — 

Draw  A'B'  at  the  intended  height,  10  ft.,  of  the  wall,  and 
produce  it  as  the  vertical  trace,  A'e',  of  a  horizontal  plane,  in 
which  is  the  quadrant  Bag,  whose  centre  is  O.  Divide  this 
quadrant  in  the  same  manner  as  EF,  as  at  a,  c,  etc.,  project  a, 
c,  etc.,  at  a',  c',  etc.,  and  as  a'g',  c'h',  etc.,  necessarily  pass 
through  O'  (being  identical  with  the  vertical  projections  of  the 
elements  of  a  cone  whose  vertex  is  00',  and  base  E7&F)  they 
are  the  vertical  projections  of  the  element  of  the  conoid. 

3°.  The  front  top  edge,  BXF  —  B'C,  is  assumed  to  be  the 
intersection  of  the  eonoidal  front-face  of  the  wall  with  a  plane, 
parallel  to  the  straight  directrix  OF,  and  having  a  slope  of  3 
to  2.  .  BF  is  then  found  by  projecting  down  B',  y',  X',  etc., 
intersections  of  B'C  with  given  elements,  upon  the  horizontal 
projections  of  the  same  elements,  as  at  B,  y,  X,  etc. 

The  back  of  the  wall,  made  of  the  given  top  thickness  AB, 


STONE-CUTTING.  95 

may  then  be  made  concentric  with  BF,  as  seen  in  plan,  by- 
making  it  tangent  to  any  sufficient  number  of  arcs,  of  radius 
AB,  and  with  their  centres  on  BXF. 

The  top  edge  of  the  bach  is  here  assumed  to  be  the  intersec- 
tion of  the  vertical  cylindrical  surface  of  the  back,  with  a  plane 
perpendicular  to  V?  and  whose  vertical  trace  is  A'C 

The  top  surface  of  the  ivall  will  thus  naturally  be  a  warped 
surface,  having  BF  —  B'F;  and  AC  —  A'C  for  directrices, 
and  H  for  its  plane  director. 

The  face-joints.  —  These  shall  be  the  elements  just  found, 
for  the  heading  joints ;  while  the  coursing  joints  shall  be  equi- 
distant horizontal  sections,  niF  —  m'T',  etc.,  which  (115)  are 
ellipses. 

The  joint  surfaces.  —  With  the  quick  curvatures  arising  from 
small  dimensions  and  large  batter,  as  in  the  present  example, 
these  surfaces  should  be  normal  to  the  face  of  the  wall  along 
the  lines  just  fixed  upon  as  the  joints  of  the  face. 

The  coursing  surfaces,  or  beds,  will  thus  be  warped  surfaces, 
which,  for  the  joint  mW  —  m'T',  for  example,  will  be  generated 
by  a  line  mA  —  m'B",  normal  to  EB  —  E'B',  and  moving 
upon  m¥  —  m'T'  as  a  directrix,  so  as  to  continue  normal  to  the 
face  of  the  wall.  Now,  since  there  can  be  but  one  tangent 
plane  at  any  one  point  of  a  surface,  and  but  one  perpendicular 
to  a  plane  at  any  one  point,  there  can  be  but  one  normal  line 
to  a  surface  at  any  one  point ;  hence,  the  warped  surface  thus 
generated  is  determinate. 

The  heading  surfaces  will  be  the  normal  hyperbolic  para- 
boloids along  the  elements  of  the  conoid  (117). 

5°.  Construction  of  the  joint  surfaces.  —  If  two  lines  are 
perpendicular  to  each  other,  and  one  of  them  be  parallel  to  a 
plane  of  projection,  their  projections  on  that  plane  will  be 
perpendicular  to  each  other.  Now  the  elements,  as  lu  —  l'u', 
of  the  conoid,  are  parallel  to  V  \  hence,  the  vertical  projections, 
r's',  n'q',  etc.,  of  the  normals  to  the  conoid,  along  lu  —  l'u', 
will  be  perpendicular  to  l'u'.  And  the  like  is  true  for  the 
other  elements.  Again,  the  normals  at  points  of  the  ellipses 
mF,  etc.,  are  perpendicular  to  the  tangents  to  those  ellipses  at 
the  same  points  ;  hence,  rs,  nq,  etc.,  oQ,  etc.,  are  perpendicular 
to  the  tangents  at  r,  n,  o,  etc.,  respectively,  to  the  ellipses 
HrF,  mriF,  etc. 

Projecting   A,   Q,  R,  etc.,  upon    the  vertical   projections, 


96  STEREOTOMY. 

ra'B",  o'Q!,  Jc'R',  etc.,  normals  at  m',  o',  Jc',  etc.,  we  have 
B"Q'R'  .  .  .  .  T,  tangent  to  m'T  at  C,T',  for  the  vertical 
projection  of  a  coursing  joint  on  the  back  of  the  wall;  and 
mAFC  — B"m'T'S'R'  as  the  normal  coursing  surface  contain- 
ing the  ellipse  m¥  —  m'F!. 

Likewise,  projecting  p,  q,  s,  etc.,  upon  the  perpendiculars  to 
Vv!  at  I',  n',  r!,  etc.,  we  find  p'q't',  a  heading  joint,  upon  the 
back  of  the  wall. 

6°.  The  tangents  at  o,  w,  r,  etc.,  to  the  ellipses  w?F,  etc.,  and 
to  which  the  normals  oQ,  etc.,  are  made  perpendicular,  may  be 
drawn  in  various  ways,  as  most  convenient  for  each  point. 

1st.  That  at  zz',  to  the  ellipse  toF,  is  here  drawn  by  the 
method  of  bisecting  the  angle  MsN  included  by  lines  from  z  to 
the  foci,  one  of  which,  /,  is  shown,  and  both  of  which  are  at 
the  intersections  of  an  arc  with  radius  Om,  and  centre  F,  with 
the  transverse  axis  mO  ;  produced  to  find  the  other  focus  fv 

2d.  That  at  n,  to  the  same  ellipse,  is  drawn  by  the  method 
of  revolution  ;  the  quadrant,  EF,  being  the  projection  of  mF 
after  a  certain  revolution  about  OF.  Then  n  appears  at  I,  and 
IG  is  the  revolved  position  of  the  required  tangent,  which,  by 
counter-revolution,  appears  at  Gn. 

3d.  The  tangent,  J>,  is  likewise  found  from  Jl,  where  the 
revolution  of  the  ellipse,  HF,  take  place  about  HO,  till  it  ap- 
pears as  a  circle  of  radius  OH. 

4th.  If  adjacent  figures  were  not  in  the  way,  so  that  other 
quadrants  of  each  ellipse  could  be  shown,  we  might  proceed  as 
follows,  by  the  method  of  conjugate  diameters.  Thus,  at  o, 
for  example,  draw  Oo  and  any  chord,  parallel  to  it,  and  the 
tangent  at  o  to  moF  would  then  be  parallel  to  the  line  from  O 
to  the  middle  point  of  this  chord,  for  such  line  would  be  the 
diameter  conjugate  to  Oo. 

7°.  Approximate  joint  surfaces.  —  With  the  considerably 
larger  dimensions,  and  less  declivity  of  face,  which  would  be 
generally  found  in  practice,  the  part  from  X  to  F  would  be 
nearly  a  vertical  plane,  and  from  BE  to  X,  nearly  a  conical, 
or  even  an  almost  vertical  cylindrical  surface.  Hence,  in  such 
a  case  the  coursing  surfaces  could  be  safely  horizontal  planes  ; 
and  the  heading  surfaces  could  be  planes.  That  through  the 
element  lu  —  l'u',  for  example,  might  be  a  plane,  determined 
by  the  element  together  with  the  horizontal  element,  at  K',  of 
the  top  surface. 


J'l  V 


STONE-CUTTING.  97 

II.  The  Directing  Instruments  and  their  Application.  As 
every  surface,  except  the  back,  which  would  generally  be  left 
rough,  of  all  the  stones  between  UU'  and  gg'  is  warped,  no 
patterns  can  be  used.  It  would  therefore  be  best  to  begin,  at 
least,  by  working  some  one  surface  by  the  method  of  squaring 
(105)  from  the  sides  of  a  circumscribing  prism,  so  far  finished 
as  to  admit  of  the  application  of  this  method. 

Having  the  front,  for  example,  thus  wrought,  beds  could  be 
made  square  with  it  by  keeping  one  arm  of  the  square  (No.  2) 
on  an  element  of  the  front,  and  the  other  on  an  element  of  the 
bed. 

With  the  often  admissible  approximate  plane  joints,  already 
described,  the  operations  would  be  much  easier,  as  patterns  of 
all  the  faces  except  the  conoidal  front  could  be  easily  found 
and  applied. 

These  general  guiding  observations,  added  to  previous  exam- 
ples, will  enable  the  student  to  construct  whatever  instruments 
may  be  necessary. 

Examples.  =  1°.  Construct  the  front  elevation  of  this  wall,  and  an  isometric, 
or  oblique  projection  of  one  of  its  stones. 

2°.  Construct  the  wall  when  OE  is  less  than  OF. 


STAIRS. 

119.  Stairs  vary  in  form  ;  first,  as  a  ivhole,  depending  on 
the  form  of  the  space  which  they  occupy  ;  second,  in  detail, 
that  is,  in  the  form  and  arrangement  of  the  separate  steps. 

Certain  practical  conditions  and  geometrical  principles  are, 
however,  common  to  all  cases  ;  hence,  a  single  example  of  a 
general  case,  fully  explained,  will  serve  as  a  standard,  from 
which  variations  may  be  made  to  any  particular  forms. 

120.  General  Geometrical  Principles.  —  All  stairs  may  be 
divided  into  two  principal  kinds. 

1st.  Straight  stairs ;  in  which  the  height,  or  rise,  and  the 
width,  or  tread,  are  each  uniform,  on  each  and  all  of  the  steps. 

2d.  Winding  stairs ;  in  which  the  rise  remains  uniform, 
while  the  tread  is  variable  at  different  points  of  each  step. 
In  this  sense,  winding  stairs  which  consist  only  of  successive 
short  flights  of  straight  stairs  running  in  different  directions, 
are  not  included. 

7 


98  STEREOTOMY. 

121.  Winding  stairs  are,  again,  of  two  species  :  — 

First,  those  which  wind  around  a  single  vertical  axis  from 
which  the  edges  of  the  steps  radiate. 

Second,  those  which  radiate  around  no  single  central  axis 
from  which  the  steps  radiate. 

In  the  former,  the  tread  is  uniform  on  a  line  of  ascent  taken 
at  any  given  distance  from  the  axis.  In  the  latter,  there  is 
but  one  such  line,  and  it  is  taken  at  that  distance  from  the 
hand-rail  which  any  one  would  naturally  choose  in  passing 
up  or  down  the  stairs,  and  may  be  called  the  line  of  passage. 

Stairs  mostly  straight  are  often  partly  winding,  at  one  or 
both  ends,  and  will  then  be  classed  under  one  or  the  other  of 
the  varieties  just  indicated. 

122.  In  winding  stairs  of  the  first  species,  the  natural  line 
of  passage  upon  them  is  obviously  a  common  or  circular  helix, 
as  in  PL  VII.,  Fig.  50 ;  where,  if  a  horizontal  and  a  vertical 
plane  be  passed  through  each  element,  they  would  evidently 
intersect  each  other  so  as  to  form  the  steps  of  such  stairs  as 
would  wind  around  the  axis  of  a  cylindrical  pit. 

In  winding  stairs  of  the  second  species,  the  horizontal  projec- 
tion of  the  line  of  passage  will  not  be  a  circle  as  in  PI.  VII. , 
Fig.  50,  but  some  other  curve,  as  LP,  Fig.  8,  better  conformed 
to  the  ground  area  covered  by  the  stairs. 

123.  Now  let  LP,  Fig.  8,  be  the  horizontal  trace  of  a  verti- 
cal cylinder,  on  which  a  point,  m,  moves  so  that  the  horizontal 
and  vertical  components  of  its  motion  are  equal,  that  is,  so  that 
if  w*%=w1w*2,  etc.,  the  heights  of  onx  above  m;  of  m2  above  mu 
etc.,  will  be  equal.  The  point  m  will  thus  generate  a  helix,  h, 
but  of  a  more  general  kind  than  the  circular  helix,  PL  VII., 
Fig.  50. 

124.  Also  if  a  horizontal  straight  line  taken  as  a  generatrix 
G,  move  upon  this  new  helix  LP,  in  the  same  manner  as  in 
PL  VII.,  Fig.  50,  the  resulting  surface  will  still  be  a  helicoid,  H, 
but  of  a  more  general  form  than  the  usual  particular  form  shown 
in  that  figure. 

But  as  the  plan,  LP,  of  h  is  no  longer  a  circle,  while  the  line 
G  continues  perpendicular  to  it  as  seen  in  plan,  Fig.  8,  the  lines 
g,  gu  g2,  etc.,  horizontal  projections  of  successive  positions  of  G, 
will  not  pass  through  any  one  point,  as  at  o  in  PL  VII.,  Fig. 
50,  but  will  intersect  each  other,  as  in  Fig.  8,  so  that  when 
g,  g\,g<i, gn,  and  thence  their  intersections,  p,  q,  r,  etc., 


STONE-CUTTING.  99 

become  consecutive,  p,  q,  r, r„,  will  form  a  curve  to 

which  g,  gv  g2,  gn will  be  tangent. 

125.  We  thus  reach  the  following  important  conclusions; 
foundation  of  the  design  of  stairs  of  every  form. 


Fig.  8. 

1°.  The  curve  pqr  is  the  horizontal  projection  of  a  vertical 
cylinder,  C,  which  replaces  the  vertical  straight  line  (axis)  o  — 
0'12'  in  PI.  VII.,  Fig.  50. 

2°.  The  curve  LP  is  the  horizontal  projection  of  a  general 
form  of  a  helix,  h,  (123). 

3°.  The  straight  line  g,  then  moves  upon  the  helix  A,  and 
remains  horizontal  and  tangent  to  the  cylinder  C.  It  thus  gen- 
erates a  general  form  of  helicoidal  surface,  H ;  such  as  forms 
the  under  surface  of  the  stairs,  and  such  as  will  contain  the 
similar  radial  edges  of  all  the  steps. 

Problem  XVI. 

Winding  stairs  on  an  irregular  ground  plan. 

I.  The  Projections.  Let  a  landing  at  AdE,  PI.  VIII. ,  Fig. 
60,  provided  for  a  door  in  the  wall  ES,  be  connected  with  a 
floor  three  feet  higher  at  CH,  by  a  flight  of  five  steps,  counting 
the  upper  level.  And  let  these  steps  be  built  three  inches  into 
the  wall,  whose  base  is  the  segment  HGFE  of  an  irregular 
polygon. 


100  STEREOTOMY. 

Let  the  line  of  passage  (121)  be  the  arc  CD  of  the  ellipse, 
whose  semi-axes,  OA,  and  OB,  are  determined  by  convenience. 

The  tread  of  each  step,  measured  from  D  on  AC,  is  12  ins. 
giving  I,  J,  K,  C.  At  these  points  draw  normals  to  the  ellipse, 
AC,  as  Dd,  etc.,  which  will  be  edges  of  steps.  On  these  nor- 
mals lay  off  18  ins.  inward  from  CD,  to  locate  the  inner  end, 
cd,  of  the  steps,  or  the  circumference  of  the  well,  as  it  is 
called. 

For  firmer  support,  let  each  step  extend  as  at  Kg,  or  Jk 
(  J2&i)  and  K^,  7  ins.  under  the  next  upper  one ;  and  then 
normals  to  CD  at  k  and  g,  will  be  the  under  edges  of  the  step 
JK,  and,  see  k2  and  g2,  in  the  general  under  surface  of  the  steps. 

This  under  surface  of  the  steps  is  a  right  helicoid,  H,  gen- 
erated by  Qkq  moving  upon  a  helix  whose  horizontal  pro- 
jection is  CD,  normal  to  it,  and  parallel  to  H  as  its  plane 
director. 

Normal  joints.  —  These,  perpendicular  on  Qq,  etc.,  to  the 
helicoid,  H,  should  (by  118)  be  hyperbolic  paraboloids.  But 
it  is  a  sufficient  approximation  to  make  them  normal  planes  at 
some  suitable  mean  point.  They  are  here  made  normal  at  g, 
k,  etc.,  points  on  the  helix  CD,  for  two  principal  reasons.  1st. 
Only  the  helix  CD  will  be  straight  in  development,  it  alone 
having  equal  arcs  for  equal  ascents  (123).  Hence,  normals  at 
its  points  can  be  more  easily  drawn.  2d.  All  the  curvatures 
are  quicker  within  than  without  CD  ;  hence  the  warped,  and 
plane  normal  surfaces,  will  differ  less  if  the  normals  be  nearer 
ed  than  to  the  wall. 

The  development.  — Proceeding  as  just  indicated,  make  C2D2 
=  CD,  and  C^  =  the  total  rise  =  3  ft.  and  C^wiH  be  the 
development  of  the  helix  over  CD,  and  which  cuts  the  front 
edges  of  the  steps. 

Next,  after  dividing  C^  into  four  equal  parts,  make  JXJ2  = 
\  0^2 ;  JiK!  =  JK  ;  J2&!  =  Jk ;  Ki<fr  —  Kg ;  and  g^g2  and  ~kx 
h2,  each,  for  example  equal  to  ^  JiJ2,  when  g2sx  parallel  to  CJ)^ 
will  be  the  development  of  that  helix,  projected  in  CD,  which 
is  in  the  helicoid  H. 

Finally,  at  g2,  k2,  etc.,  draw  g2fi,  hh,  etc.,  perpendicular  to 
C1D1,  and  they  will  be  lines  of  the  required  normal  planes  to 
H,  at  g,  k,  etc.  Do  the  same  for  the  other  steps,  and  the  figure 
EMijfeJa  will  be  the  complete  development  of  the  section  of  the 
steps  made  by  the  vertical  cylinder  whose  base  is  CD. 


STONE-CUTTING.  101 

Then,  to  complete  the  plan,  make  gf  =  Teh  =  g\fx,  etc.,  and 
the  lines  as  Mhm,  parallel  to  Qkq,  etc.,  will  be  the  traces  of 
the  normal  planes  to  H  upon  the  treads. 

II.  The  directing  instruments,  besides  Nos.  1  and  2,  are 
these :  — 

No.  8,  the  pattern,  Mhrnl,  of  the  top  of  a  step. 

No.  4,  the  pattern,  M'L'B/,  of  the  wall  end  of  a  step.  This 
is  found  by  projecting  the  outer  end,  QL,  of  a  step  upon  a  par- 
allel plane,  as  shown  ;  making  all  the  .vertical  distances  equal 
to  the  like  ones  on  the  development ;  Q'Qi  =  gxg2 ;  P'Pi  = 
KiK.  (greater  than  JiJ2,  since  K^fr  is  greater  than  k'k{). 

No.  5,  the  pattern,  n^bi,  of  the  inner  end  of  a  step.  This, 
for  the  step  EcZrR,  is  the  development  of  its  inner  end.  Then 
rxnx ;  rx\  ;  rxdx,  etc.,  equal  rn,  rl,  rd,  etc.,  on  the  plan ;  and  the 
heights,  h}2,  etc.,  equal  those  on  the  delopment  KjKg,  etc. 

No.  6  is  a  normal   joint  bevel,  giving  the  constant  angle 

III.  Application.  —  Having  chosen  a  sufficient  block,  bring 
its  intended  top  to  a  plane,  and  mark  its  form  by  No.  3. 
Work  the  rise  and  the  end,  square  with  the  top,  using  Nos. 
4  and  5  to  give  the  forms  of  the  ends.  Work  the  normal 
joints  by  No.  6,  and  the  helicoidal  under  side  by  No.  1,  ap- 
plied on  points  transferred  from  the  drawing,  where  elements 
would  meet  ql  and  QL,  for  the  step  I  J,  for  example. 

Other  Forms  of  Stairs. 

126.  Other  stairs  are,  for  want  of  space,  merely  suggested  by 
the  steps,  illustrated  in  Figs.  61,  62,  which  are  both  adapted  to 
circular  stairs  ;  that  is,  those  placed  in  a  cylindrical  case.  They 
are  contrasted  in  the  manner  of  support.  In  Fig.  61,  which 
shows  a  plan,  elevation  of  the  back  edges,  and  a  development 
of  the  cylindrical  outer  end  BD,  the  central  open  cylinder,  or 
well,  is  filled  by  a  core,  composed  of  the  cylindrical  wings,  or 
ears,  O,  solid  with  the  inner  end  of  each  step.  The  core  is 
sometimes  larger,  and  then  solid,  and  with  the  steps  indented 
into  it,  as  at  the  outer  ends  in  Fig.  60. 

Fig.  62  is  an  oblique  projection  of  one  step  of  circular  stair 
with  an  open  well ;   and  the  steps  are  supported  by  an  ear  CE 


102 


STEREOTOMY. 


at  the  outer  end  whose  whole  thickness,  Fe,  is  indented  into  the 
wall,  giving  them  a  wide  horizontal  support. 

Examples.  —  1°.  Observing  that  the  curves,  CDA  and  XY,  PI.  VIII.,  Fig.  60, 
have  the  relation  of  involute  and  evolute  to  each  other,  represent,  with  the  pat- 
terns, stairs  in  which  XY  shall  be  assumed. 

2°.  Stairs  in  which  cd  shall  be  assumed. 

3°.  Construct  stairs  whose  steps  shall  be  formed  as  indicated  in  Fig.  62. 

4°.  Construct  circular  stairs  in  a  cylindrical  case,  with  a  central  post  formed  of 
steps  like  that  of  Fig.  61. 

5°.  In  a  flight  of  five,  or  more,  steps  against  one  reach  of  wall,  as  GF,  Fig.  60, 
construct  the  intersection  of  that  wall  with  the  helicoid,  H,  of  the  under  surface  of 
the  stairs. 


CLASS  IV. 
Structures  containing  Double-Curved  Surfaces. 

Problem:  XVII. 

A  trumpet  bracket  with  basin  and  niche . 

I.  The  Projections.  These  are  a  plan,  RrE  ;  a  front  eleva- 
tion A'B'E"  ;  and  a  sectional  side  elevation,  0"'R"E"". 

1°.  The  outlines  of  the  bracket.  Rrrx  PI.  IX.  Fig.  63,  is  a 
wall  of  the  general  thickness  QK ;  but,  where  the  bracket  is 
attached,  of  the  additional  thickness,  rxs.  The  front  of  the 
bracket  is  composed  of  two  cylindrical  surfaces ;  one  vertical, 
with  the  radius  OA ;  the  other  horizontal,  with  its  elements, 
AB,  na,  etc.,  parallel  to  the  ground  line  A'B'. 

The  form  of  the  latter  cylinder  is  made  to  depend  on  the 
given  curve,  AEB  —  A'E'B',  of  their  intersection,  where  A'E'B' 
is  a  semicircle.  This  curve  would  be  the  intersection  of  the 
vertical  cylinder,  with  a  cylinder  of  revolution  whose  vertical 
projection  would  be  A'E'B',  as  shown  more  clearly  in  the  aux- 
iliary Fig.  64,  where  AEB  —  A'E'B'  is  the  intersection  of  the 
vertical  cylinder,  CD  — CD',  with  the  cylinder,  UT  — A'E'B', 
which  is  perpendicular  to  the  vertical  plane.  By  Theorem  I., 
the  projection,  E'"0"',  of  the  intersection  AEB — A'E'B',  on  a 
plane,  as  0"'X,  parallel  to  the  plane,  EF,  of  the  axes  of  the  cyl- 
inders of  revolution,  is  a  hyperbola.  The  vertical  cylinder  is 
then  cut  away  to  this  hyperbolic  profile,  E'"0'" ;  so  that  the 
face  of  the  bracket  within  A'E'B'  is  a  cylinder  parallel  to  the 
ground  line  and  with  a  hyperbolic  right  section,  or  base  O'E' — 
0"'E'". 

The  top  of  the  bracket  is  a  plane  annular  surface,  between 
AEB  — A"B"  and  the  circular  edge,  of  radius  OH,  of  the  hem- 
ispherical basin,  H'G'I'  —  J'G"J". 

2°.  The  joints  of  the  bracket.  —  Having  found  0"'E'",  as  may 
be  seen  by  inspection,  divide  A'E'B'  into  equal  parts,  here  five, 
and  draw  the  radial  joints,  as  A'O';  c'O',  etc.,  limited,  to  avoid 
thin  edges,  by  a  semi-cylindrical  stone  of  radius  UT —  O'T',  = 
R'V".     Then  — 


104  STEREOTOMY. 

To  find  intermediate  points  in  the  joints  on  the  vertical  cylin- 
der.—  Assume  b'  as  such  a  point.  Its  horizontal  projection  is 
b,  and  auxiliary  projection  J1?  which  by  revolution  appears  at  b", 
intersection  of  b2b"  and  b'b". 

To  find  intermediate  points  in  the  joints  on  the  horizontal  cyl- 
inder.—  Here  it  is  the  horizontal  projection  of  the  point  that 
needs  construction.  Assuming  d',  for  example,  project  it  at 
d",  on  0"'E',  thence  to  sX  and  revolve  upon  Xm;  whence  pro- 
ject upon  d'd  at  d.  Or,  by  the  method  of  transference,  make 
dxd  =  d"'d"  (50). 

By  the  same  method,  applied  to  b",  make  b'"b"  =  ob,  to  find 
b".  Also,  make  h"e'"  =  hh!",  to  find  h".  In  similar  ways  the 
points  of  all  the  joints  of  the  bracket  can  be  found ;  as  at 
e'"h'{  =  h'"hx,  in  finding  Al5  a  point  of   the  basin  joint,  hxgk 

—  h'g'—Kfh'i'. 

The  horizontal  projections,  as  h^gk,  of  the  basin  joints,  above 
gg',  on  the  vertical  semicircle,  H'GT,  of  the  basin,  are  found 
by  horizontal  circles,  as  that  with  radius  j'V,  each  of  which  will 
contain  four  points  of  the  two  basin  joints,  of  which  points  i$ 
is  one. 

3°.   The  Niche.  —  The  niche  is  a  vertical  semi-cylinder,  CFD 

—  C'C'D'D",  covered  by  the  quarter  sphere  included  between 
the  horizontal  semicircle,  CFD  —  CD",  and  the  vertical  one, 
CD  —  C"E"D".  The  joints  of  the  spherical  part  are  circular, 
and  in  planes  which  radiate  from  the  diameter,  OF  —  O"  — 
0""F"  ;  and  are  limited  by  the  cylindrical  stone,  0"L/P". 

Intermediate  points  of  these  joints,  as  N',  are  readily  found 
on  the  plan  and  side  elevation,  in  various  ways.     Thus,  MMX 

—  M'N'Mi  —  M"N"  is  a  vertical  circle  through  W,  which  point 
is  thence  projected  upon  MM^  at  N,  and  upon  M/rN"  at  N". 
This  circle  might  have  been  made  horizontal  through  N' ;  or 
the  plane  of  the  joint  might  have  been  revolved  about  OK  — 
0"K'  as  an  axis,  when  the  circular  joint  would  have  fallen  on 
K'E"  and  W  at  n".  Then  make  r^N  =  n"W,  which  will  give 
N,  as  before. 

Varied  constructions  are  useful,  in  case  some  one  of  them 
does  not  conveniently  apply  to  certain  points.  Points,  as  K', 
being  projected  at  K,  and  K",  and  L/  at  L,  and  L",  KNL  and 
K"N"L"are  the  other  two  projections  of  the  circular  joiDt  K'L\ 

II.   The  Directing  Instruments.  —  These  will  consist  of  pat- 


PIT! 


I 


STONE-CUTTING.  105 

terns  of  the  plane  and  developable  faces  of  all  the  stones,  with 
a  sufficient  number  of  bevels  to  insure  the  correct  relative  po- 
sitions of  all  the  surfaces. 

After  the  numerous  preceding  examples  of  developing  the 
convex  surfaces  of  stones  of  irregular  shape,  the  construction  of 
the  required  patterns  can  be  made  from  the  following  general 
directions. 

Take  the  stone,  A" A'  VW,  of  the  bracket,  and  L'K'p'  of  the 
niche,  as  being  the  most  irregular  ones.  Let  each  extend  to  the 
vertical  plane  back,  Br  —  R/'Q",  of  the  wall.  Develop  the 
entire  convex  surface,  both  lateral  and  end  faces,  of  each,  as  in 
previous  examples.  The  following,  including  the  necessary 
bevels,  will  thus  be  found. 

No.  1,  the  straight  edge. 

No.  2,  the  square.     Then  for  the  bracket  stone  alone  — 

No.  3,  the  pattern  of  the  back,  =  A"h'Y'v't'. 

No.  4,  that  of  the  top,  =  A'"A2A3HA1/iA. 

No.  5,  that  of  the  radial  joint,  hiVuh2h3ghl  — Y'h'. 

No.  6,  that  of  the  opposite  radial  joint,  AA'"v1«». 

No.  7,  that  of  the  surface,  Yvvxu. 

No.  8,  the  development  of  the  vertical  cylindrical  surface, 
AM  —  A"h'i't'. 

No.  9,  the  development  of  the  portion,  nivY — n'i'v'Y'  —  i"f", 
of  the  horizontal  hyperbolic  cylinder  of  the  front. 

Nos.  10  and  11,  as  shown  on  the  figure. 

The  corresponding  guides  for  the  niche  stone  can  readily  be 
found. 

To  avoid  the  too  acute  angle  at  hx  in  the  bracket  stone,  H'G'I' 
might  have  been  an  arc  of  120°  or  less  ;  or  the  portion  of  the 
bracket  above  E'  might  have  been  a  single  stone  thick  enough 
to  contain  the  basin. 

III.  The  Application.  This  is,  in  the  main,  sufficiently  ob- 
vious, from  the  description  of  the  patterns,  and  the  previous 
essentially  similar  cases.  The  back  of  both  the  bracket  and 
the  niche  stone  may  properly  be  wrought  first,  since  all  the  lat- 
eral faces  are  perpendicular  to  it.  The  order  and  manner  of 
using  the  remaining  guides  may  be  left  to  the  workman. 

Examples.  —  1°.  Construct  the  niche  alone. 

2°.  Construct  the  bracket  alone,  and  without  the  basin. 


106 


STEREOTOMY. 


Theorem  III. 

The  conic  section  whose  principal  vertex  and  point  of  contact 
with  a  known  tangent  are  given,  will  be  a  parabola,  ellipse,  or 
hyperbola ;  according  as  the  given  vertex  bisects  the  sub- 
tangent,  or  makes  its  greater  segment  without,  or  within,  the 
curve. 

In  both  the  ellipse  and  the  hyperbola,  referred  to  their  cen- 
tres and  axes,  the  subtangent  is  a  fourth  proportional  to  the 
abscissa  of  contact,  and  those  segments  of  the  transverse  axis 
which  meet  on  the  ordinate  of  contact.  That  is,  in  Figs.  9 
and  10,  CO  :  Oa  : :  OA  :  OT  ; 

whence,  by  division,      CO  :  Oa  —  CO  : :  OA  :  OT— OA  ; 
or  CO  :  CA  : :  OA  :  TA 


Fig.  9, 


C     T 


I 


Fig.  10. 


Now,  always,  in  the  ellipse,  CA>CO 

.-.  TA>OA 

But  in  the  hyperbola,  CA<CO 

.-.  TA<OA. 

In  the  parabola,  AO  =  AT. 


STONE-CUTTING.  10T 

Problem  XVII. 

The  hooded  portal. 

The  Projections.  —  PL  IX.,  Fig.  65.  This  construction  is 
known  in  France  as  the  recessed  gate  of  St.  Anthony,  it  being 
that  gate  near  to  the  Bastile,  which  led  to  the  suburb  called  St. 
Anthony.  The  figure  represents  a  broken  plan  and  front  ele- 
vation, and  a  vertical  section  through  the  axis  of  the  portal. 
Its  design  is  to  give  to  a  portal,  closed  by  rectangular  gates, 
something  of  the  grander  effect  of  a  semicircular  topped  portal, 
closed  by  gates  of  like  form,  as  in  PI.  VI.,  Fig.  45. 

Let  AcFmVO  be  half  of  a  horizontal  section  of  the  portal, 
below  its  top,  V'F'.  Then,  as  in  Problem  XI.,  the  entire 
passage  embraces  three  parts,  of  which  one  half  of  each  is  as 
follows  :  the  portal  proper,  EVmF ;  the  gate  recess,  ~Ec"ce ; 
and  the  converging  embrasure,  OecA,  flanked  by  the  jambs, 
one  of  which  is  Ac. 

The  tops  of  the  two  former  parts  are  horizontal  planes.  The 
latter  is  covered  by  the  very  peculiar  double-curved  surface, 
characteristic  of  the  structure,  and  generated  by  a  variable 
semi-ellipse  ;  which,  starting  from  the  semicircle  of  radius  O A, 
as  its  initial  position,  moves  so  that  its  centre  shall  remain  on 
Oe  —  O',  one  vertex  upon  Ac,  and  another  upon  a  fixed  ellipse, 
C"p"e",  whose  axes  are  O  —  O'C  and  Oe  —  O'.  Hence,  this 
elliptical  generatrix  varies  from  the  semicircle  of  radius  OA  to 
the  straight  line  2ec.  Thus,  when  this  ellipse  has  reached  the 
plane  ap,  its  semi-axes  will  be  ap  and  O'p',  =  qp".  Likewise, 
bf  and  O'f  are  the  semi-axes  of  another  position  of  the  movable 
ellipse. 

We  may  note  in  passing  that,  beginning  at  the  vertical 
plane  OA,  the  horizontal  axis  will  diminish  more  rapidly  than 
the  vertical  one,  until  we  reach  the  point  of  contact  of  that 
tangent  to  c"p"e",  which  makes  the  same  angle  with  0"D" 
that  Ac  does  with  OA.  Hence,  as  O'A'C  is  a  circle,  a  few 
positions  of  the  movable  ellipse  near  it  will  have  their  longer 
semi-axes  vertical. 

The  joints,  as  L'J,  in  the  vertical  plane  portion  exterior  to 
the  recess,"  are  straight,  and  radial  to  the  semicircle  of  ra- 
dius O'A'. 

The  face  joints,  as  L'M7,  within  the  recess,  are  best  made 


108  STEREOTOMY. 

normal  to  both  of  the  limiting  positions  of  the  variable  ellip- 
tic generatrix.  This  result  may  be  obtained  by  making  these 
joints,  as  seen  in  front  elevation,  as  circular  arcs,  tangent,  as  at 
I/,  to  the  radial  joints,  L' J,  etc.,  just  described,  and  with  their 
centres  on  O'A'.  But  such  joints  will  divide  20'c'  unequally. 
If  the  latter  result  be  thought  undesirable,  20V,  the  top  line 
of  the  gate  recess,  may  be  divided  equally,  as  in  the  figure,  and 
the  joints  may  be  either  conic  sections  or  curves  of  two  centres, 
tangent,  as  at  L',  to  the  radial  joints,  and,  as  at  M',  to  vertical 
lines. 

The  construction  is  illustrated  in  the  joint  K'G',  which 
(Theor.  III.)  is  elliptical,  since  the  greater  segment,  K'O',  of 
the  subtangent,  O'o,  is  exterior  to  the  curve.  Bisecting  the 
chord  G'K',  and  joining  its  middle  point,  n',  with  d!,  the  inter- 
section of  the  tangents  at  G'  and  K',  we  have,  by  a  property  of 
the  ellipse,  d'n'  as  a  diameter  ;  which  therefore  meets  the  axis, 
O'A'  (and  the  opposite  symmetrical  diameter),  at  X,  the  centre 
of  the  curve ;  whence  the  shorter  axis  can  be  found  from  the 
property  of  the  subnormal,  oS,  expressed  by  the  proportion, 

Xo  :  oS  : :  a2  :  52, 
where  a  is  the  semi-transverse  axis,  XK'  ;  and  5,  which  equals 
the  semi-conjugate  axis,  =  AB,  Fig.  66,  is  found  by  the  usual 
construction  from  elementary  geometry. 

Examining  the  joints  M'L'  and  P'Q',  we  find  M'0'>*  i'h', 
but  P'O'  <  PV.  Hence,  by  Theor.  III.,  the  joint  %1'V 
should  be  elliptical,  and  P'Q'  hyperbolic ;  but  the  differt  *ces, 
M'O' —  M'A'  and  P'O'  — PV,  are  so  small,  that  they  are  here 
made  with  sufficient  accuracy  as  circular  arcs,  whose  centres 
are  on  O'X. 

The  horizontal  projections  of  the  joints  are  found  by  pro- 
jecting down  their  intersections  with  the  contours  of  the  sur- 
face, made  by  the  vertical  planes,  pa  and  /A,  as  is  fully  shown 
for  the  joint  K'I'H'G' ;  whose  horizontal  projection  is  KIHG. 

II.  The  Directing  Instruments.  —  Most  of  these  can  be 
sufficiently  indicated  by  a  description  of  the  most  irregular 
stone  of  the  structure;  that  whose  vertical  projection  is 
RYy  F'K'G',  and  which  is  more  clearly  exhibited  in  the  oblique 
projection,  Fig.  67,  like  points  having  like  letters  in  both 
figures. 

The  many  surfaces  of  this  stone  are :  — 


STONE-CUTTING.  109 

1°.  The  vertical  rectangular  plane  side,  Yz/UY'. 

2°.  The  vertical  plane  back,  Y'Uk. 

3°.  The  vertical  plane  front,  GRY?/Z. 

4°.  The  horizontal  plane  base,  ~UkK"c"zZy. 

5°,  6°.  The  horizontal  plane  top,  RYR'Y' ;  and  small  hori- 
zontal plane  surface,  Kcc'K'. 

7°,  8°,  9°,  10°.  The  four  minor  vertical  plane  faces,  K'K"&, 
K'K' W,  czc",  and  AZz,  respectively,  in  the  portal,  gate  recess, 
and  jamb. 

11°.  The  oblique  plane  surface,  GRG'R'. 

12°.  The  elliptic  cylindrical  surface,  G'k'KRG.  Fig.  68,  is 
the  development  of  the  like  surface  on  M'L',  joined  with  the 
plane  portion  on  L/J. 

13°.  The  double-curved  surface,  AcKHG. 

The  last  surface  being  non-developable,  no  pattern  of  it  can 
be  made,  but  templets  fitted  to  any  of  its  vertical  sections, 
parallel  to  C"e",  or  to  f'b',  or  to  its  horizontal  sections,  can  be 
made. 

These  templets,  with  patterns,  easily  made,  of  the  other  sur- 
faces, and  the  square  and  straight  edge,  will  be  ample  guides  in 
working  this  stone. 

III.  Application.  —  First  form  the  surface,  YY'Uy,  it  being 
the  largest  and  simplest ;  next  the  back  ;  and  then  the  base 
and  front,  and  all  the  other  plane  surfaces,  each  of  which  is 
square  with  one  or  more  of  the  others. 

The  cylindrical  surface,  GG'Kk',  may  then  be  wrought 
square  with  the  back  upon  G'k',  as  a  given  edge,  or  directrix, 
previously  found  by  the  pattern  of  the  back.  Or  it  may  be 
wrought  by  templets  fitted  to  the  profile,  R'G'k'.  Its  edges 
may  then  be  scored  on  the  stone  by  a  pattern  corresponding 
to  that  of    the    cylindrical  joint  on   M'L/,  shown  in  Fig.  68. 

These  operations  will  give  all  the  bounding  edges  of  the 
one  remaining  surface,  which  is  double-curved.  After  approx- 
imately hewing  out  this  portion  of  each  of  the  stones,  they  can 
be  accurately  put  in  place,  since  all  the  other  surfaces  of  each 
will  have  been  previously  completed.  The  total  double-curved 
surface  of  the  recess  can  then  be  wrought  at  once,  by  means  of 
the  templets,  last  described  in  the  list  of  guiding  instruments. 

Examples.  —  1°.  Construct  the  figure  with  two  centred  joints  in  the  front 
elevation. 


110  STEEEOTOMY. 

2°.  Make  an  isometrical  or  an  oblique  projection  showing  the  under  side  of 
the  stone  shown  in  Fig.  67. 

3°.  Make  like  projections  of  the  stone  M'JTR. 


Problem  XIX. 
An  oblique  lunette  in  a  spherical  dome. 

I.  The  Projections.  —  A  lunette  is  formed  by  the  intersection 
of  two  arched  spaces,  both  of  stone,  and  of  unequal  heights,  so 
that  the  groin  curves  will  be  of  double  curvature. 

1°.  Arrangement  of  projections.  —  PL  X.,  Fig.  69.  These 
are  a  plan,  and  two  elevations,  on  two  vertical  planes,  V  and  Vu 
at  right  angles  to  each  other,  and  whose  ground  lines  are  respec- 
tively O'X  and  0"X.  As  in  all  similar  cases,  the  projections 
of  any  point  on  V  and  on  Vi>  will  then  be  at  equal  heights 
above  O'X  and  0"X. 

Given  parts  and  dimensions.  —  In  the  plan,  the  circles,  OA 
of  11  ft.  radius,  and  OH  of  13' :  6"  radius,  are  the  horizontal 
traces  of  the  interior  and  exterior  surfaces  of  the  dome.  The 
former  is  a  hemisphere  ;  the  latter,  partly  cylindrical,  as  indi- 
cated in  the  section  shown  on  the  plane  0"X,  is  there  gener- 
ated by  H'H",  6':  10"  high.  The  radius,  D"x,  of  the  extrados, 
is  16  ft.,  where  #  is  3  ft.  below  the  centre,  O",  of  the  intrados. 
The  elevation  on  O'X  shows  a  right  section  of  the  arch,  its 
inner  radius  4  ft.,  its  outer  one  8  ft.,  its  thickness  at  the 
crown  1' :  6"  ;  and  the  perpendicular  distance  of  its  axis,  oxo', 
from  the  diameter,  HO,  6' :  3". 

From  these  data  all  the  remaining  constructions  are  made. 

2°.  The  groin.  —  Any  horizontal  plane  will  cut  a  horizontal 
circle  from  the  sphere,  and  two  elements  from  the  arch,  which 
will  meet  that  circle  in  two  points  of  the  groin.  Thus,  the 
plane,  a'lm(m1v'),  cuts  from  the  sphere  the  circle  of  radius 
Oax  (=  vy)  and  from  the  arch  the  two  elements,  of  which  one 
at  a[,  being  projected  on  H,  intersects  circle  Oax  at  ax,  as  shown, 
and  thence  gives  its  side  elevation  ax  on  vy.  Other  points 
being  found  in  the  same  manner,  give  the  groin  curve*  aca5  — 
a'c'a'5  —  a"c"a's. 

3°.  The  horizontal  projection  of  the  groin  is  an  arc  of  a 
parabola.  —  To  prove  this,  refer  the  intrados  of  the  sphere  and 
cylinder  to  the  three  rectangular  coordinate  axes  :  OHl5  as  the 
axis  of  X  ;    OH,  as  the  axis  of  Y ;  and  the  vertical  at  O,  as 


STONE-CUTTING.  Ill 

the  axis  of  Z.  Then,  neglecting  the  usual  negative  sign  of  or- 
dinates  to  the  left  of  the  origin,  O,  as  not  relating  to  the  form 
of  the  line  sought,  we  have  for  the  point  a^f,  for  example, 

(OA)2  +  (ha,)2  +  (A'O2  =  R2 
where  R  =  the  radius,  OA,  of  the  sphere. 

That  is  z2  +  #2-f-z2  =  R2.  (1) 

And  as  the  like  is'  true  for  every  point  of  the  sphere,  (1)  is 
called  the  equation  of  the  sphere,  referred  to  its  centre. 

Again,  (V  A')2  +  (h'a'^f  =  (Vai)2  =  r2. 

That  is,  calling  O'o'—a, 

(a  —  xy  +  z2  =  r*.  (2) 

and  as  the  like  is  true  for  every  point  of  the  cylinder,  this  is 
called  the  equation  of  the  cylinder  for  the  given  axes  of 
reference. 

Now  that  points,  as  axa'v  may  be  common  to  both  surfaces, 
and  hence  be  points  of  their  intersection,  the  x,  y,  and  z  of  (1) 
and  (2)  must  be  the  same.  That  is,  (1)  and  (2)  will  both  be 
true  at  once  for  the  same  point,  so  that  we  can  substitute  any 
term  in  one  for  the  like  term  in  the  other. 

Then,  from  (1) ,         y1  —  R2  —  (z2  +  z2) 
and  from  (2),      (x  -f-  22)  =  (r2  -J-  2a#  —  a2) 
whence,  y2  =  —  lax  -f-  (R2  —  r2  -f-  a  )  (3) 

which,  since  z  is  eliminated,  is  the  equation  of  the  curve  aa5,  in 
the  plane  XY.  Also,  the  term  in  the  parenthesis  is  constant, 
being  made  up  of  constants,  and  as  a  is  a  part  of  it,  it  may  be 
written  2wa,  and  (3)  then  becomes 

y2  =  —  2ax  -f-  2na  =  —  2a(x  —  ri)  (4). 

If  now  we  shift  the  origin  O  to  the  left,  on  the  axis  of  X,  so 
as  to  make  x  =  x-\-ni 

(4)  will  become,  y2=  —  2ax  (5). 

Restoring  now  the  neglected  sign  of  x,  we  finally  have 

y2  =  2ax  (6) 

the  usual  form  of  the  equation  of  a  parabola  lying,  as  a5a  does, 
to  the  right  of  its  vertex  taken  as  the  origin.  The  curve  aa5  is, 
therefore  the  arc  of  a  parabola,  of  which  HO  is  the  axis. 

4°.  Joint-lines  and  surfaces  of  the  sphere.  —  The  coursing 
joints  on  that  part  of  the  sphere  which  is  independent  of 
the   lunette,  are   horizontal  circles,  IQ  —  I"Q",  GR  —  G"R", 


112  STEREOTOMY. 

etc.,  found  by  dividing  the  meridian  of  radius  0"A"  into  an  odd 
number  of  equal  parts,  —  here  eleven.  The  broken  joints, 
RQ  —  R"Q",  etc.,  are  arcs  of  meridians. 

The  beds  of  the  dome  voussoirs  are  the  conical  surfaces,  as 
P"Q"I"  J",  having  the  centre,  00",  of  the  sphere  for  a  com- 
mon vertex,  and  intersecting  the  spherical  surfaces  in  the 
horizontal  circles,  as  I"Q"  and  J"P'\ 

5°.  Radial  joint-surfaces  of  the  lunette.  —  These  are  wholly 
plane,  and  their  edges  are  the  intersections  of  these  planes 
with  the  several  surfaces  of  the  dome  and  arch. 

Divide  the  arc  a'c'a\  so  that  a[  shall  he  lower  than  E",  the 
corresponding  first  one  from  A",  of  the  eleven  equal  divis- 
ions, on  the  dome  section ;  here,  into  five  equal  parts.  The 
reason  for  this  will  soon  appear. 

The  lunette  joints  in  the  spherical  intrados. — -lb2  is  the  trace 
on  V  of  the  plane,  R%,  of  the  horizontal  circle  GR  —  G"R". 
This  plane  cuts  the  plane  of  the  joint  o'd2  in  a  horizontal  line 
at  b'2,  which,  by  projection,  gives  b2,  and  thence  b2.  Hence, 
a2b2 —  a'2b2 — a 2 b'2  is  one  oi  these  joints,  showing  that  ax  must 
be  lower  than  E",  in  order  that  there  should  be  such  a  joint. 
The  others  are  found  in  the  same  way. 

6°.  The  lunette  joints  in  the  conical  beds  of  the  dome. —  One 
of  these  is  the  intersection  of  the  plane  o'd2,  with  the  conical 
bed,  C"G"R"R'".  To  find  it,  draw  qd'2,  at  the  height  of  C"qi, 
to  give  qd2,  the  trace  of  the  horizontal  plane  of  Cd2  —  C"qv 
upon  V»  As  before,  this  plane  cuts  from  the  plane  o'd2  a  per- 
pendicular to  V  at  d'2,  which,  in  horizontal  projection,  gives  d2, 
on  the  horizontal  projection,  Cd2  (C  being  projected  from  C"), 
of  the  circle  considered  ;  and  thence  d2.  The  joint  sought  is 
evidently  a  hyperbola,  it  being  the  intersection  of  the '  plane 
oxo]d'2  with  the  cone  whose  axis  is  the  vertical  at  O,  and  whose 
slant  is  that  of  G"0".  Hence,  make  p'O'o' =  G"0"A",  and 
Op  —  O'p'  is  that  element  whose  intersection  with  the  plane, 
0]p'd2,  is  the  vertex, p'p,  of  this  hyperbola;  whose  horizontal 
projection  pb2d2  can  now  be  more  accurately  drawn  than  with- 
out the  aid  of  the  vertex  p.  Finally,  make  O"/?"— the  height 
of  p',  and  p"b2d2  is  the  vertical  projection  of  the  same  hyper- 
bolic joint,  of  which  only  b2d?  —  b'2d2  —  b2d2  is  real. 

Any  other  hyperbolic  joints  are  found  in  the  same  way. 

Other  lunette  lines.  —  These  are,  for  the  same  joint  plane 
o'd'2,  the  circular  arc  d2e  —  d2e' — d2e",  on  the  spherical  extrados  • 


pi.vir 


Dr'-4*_ 


STONE-CUTTING.  113 

ef2  —  e',  on  the  horizontal  ledge,  generated  by  D"H"  ;  f2g2  — 
e'g'2  on  the  cylindrical  back  of  the  dome  ;  g2ux  —  9i->  on  the 
extrados  of  the  arch;  uxu — g2d2,  a  radial  edge  in  the  arch; 
and  ua2  —  d2,  on  the  intrados  of  the  arch. 

'  The  large  diameter  of  the  arch,  as  compared  with  the  radius 
OAx,  carries  the  point  tt't"  nearly  out  of  the  quadrant,  OAAx, 
unless,  as  shown  at  ft",  it  be  taken  lower  than  the  correspond- 
ing point  bb\. 

II.  The  directing  Instruments. —  These,  besides  Nos.  1  and 
2,  are  patterns  of  all  the  plane,  cylindrical,  and  conical  surfaces 
of  voussoirs,  with  certain  bevels,  as  follows,  taking  for  illustra- 
tions the  stone  between  o'd'2  and  o'g{  of  the  lunette,  and  the 
stone  R"Q"S"T",  of  the  dome. 

No.  3  shows  the  real  form,  a\g']a'2g2,  of  the  plane  end  of  the 
lunette  stone,  which  is  in  the  plane  V. 

Nos.  4,  5,  and  6,  Fig.  70,  are  patterns  of  the  intrados  (No. 
5)  of  the  same  stone,  and  of  the  two  radial  plane  joints  when 
folded  into  the  paper.  Their  construction  is  obvious,  since  like 
points  have  like  letters  with  Fig.  69,  and  are  found  by  ordi- 
nates  from  the  vertical  plane  end,  No.  3,  in  the  plane  V- 

Useful  bevels  (not  shown),  would  be  No.  7,  giving  the  angle 
h"WW;  and  Nos.  8  and  9,  giving  the  positions  of  the  plane 
beds  on  d2d2  and  d^g\,  relative  to  the  intrados  dxd2. 

No.  10,  the  pattern  of  that  plane  end,  MNW,  of  this  stone, 
which  is  in  the  dome,  is  G"C"D"H"&"B"E". 

From  MN  to  a2b2  is  a  spherical  zone. 

From  N5  to  bxdx  is  a  conical  zone,  No.  11. 

From  5W  in  the  plane  Wk"  to  dxfx,  is  a  horizontal  plane 
surface,  No.  12. 

From  the  vertical  line,  W  —  k"H",  extends  the  vertical  cyl- 
indrical back,  No.  13,  of  the  dome,  intersected  by  those  sur- 
faces of  the  lunette  which  are  parallel  to  its  axis  o'o^ 

The  three  remaining  surfaces  of  the  part  of  the  stone  in  the 
dome,  are  the  plane  annular  portion,  No.  14,  generated  by 
D"H";  the  spherical  portion  generated  by  D"C",  and  a  conical 
portion,  No.  15,  generated  by  C"G";  all  starting  from  the 
plane  OW,  and  all  limited  at  their  intersections  with  that 
portion  of  the  stone  which  is  in  the  arch. 

Patterns  of  these  surfaces,  so  far  as  developable,  may  readily 


114  STEREOTOMY. 

be  made  ;  also  bevels,  conveniently  giving  the  position  of  their 
horizontal  edges,  relative  to  the  end  in  the  plane  OW. 

Thus  this  very  irregular  stone  has  thirteen  faces,  plane, 
cylindrical,  conical,  and  spherical. 

To  gain  as  full  an  idea  of  it  as  drawings  alone  can  give,  com- 
plete its  projection  on  V  ;  and  make  two  or  more  isometrical, 
or  oblique  projections  of  it. 

For  the  proposed  stone  of  the  dome,  the  pattern,  No.  16, 
C"G"I"J",  of  its  vertical  plane  end  will  be  needed  ;  and  those 
of  its  conical  beds,  as  P"Q"V"S",  Nos.  17  and  18. 

No.  17,  for  example,  Fig.  71,  is  the  development  of 
R"R"/U'/T",  found  by  describing  the  arcs  from  O,  with  radii 
equal  to  0"G"  and  6"C",  and  by  making  R"T"  =  RT  from 
the  plan. 

Finally,  bevels  like  No.  19,  will  be  useful,  giving  the  rela- 
tive positions  of  elements  of  the  conical  beds,  and  great  circles 
of  either  the  intrados  or  extrados  of  the  dome.  And  a  tem- 
plet, No.  20,  should  be  cut  to  an  arc  of  a  great  circle  of  the 
spherical  intrados.  \ 

III.  Application.  —  For  a  stone  as  irregular  as  that  of  the 
lunette,  the  method  by  squaring  (105)  is  preferable,  if  not  in- 
dispensable. Then  form  a  right  prism,  the  pattern  of  whose 
base  shall  be  the  horizontal  projection,  ^^WM,  of  this 
stone ;  and  upon  whose  rear  and  lateral  faces  the  two  plane 
heads  can  be  marked  by  Nos.  3  and  16. 

Next,  the  intrados  and  plane  joints  of  the  arch  portion  of  the 
stone  can  readily  be  made  square  with  the  back  by  No.  2,  and 
formed  by  Nos.  4,  5,  and  6. 

The  plane,  W/^5,  is  readily  made  ;  square  with  the  end 
on  MW,  and  marked  by  No.  12  ;  the  cylindrical  back,  square 
with  the  last  surface,  and  marked  by  No.  13 ;  and  the  spher- 
ical surface,  MNa1aJ>2,  by  Nos.  19  and  20. 

Pendentives. 

127.  In  connection  with  domes,  the  related  subject  of  square 
areas,  covered  by  spherical  surfaces,  may  be  noticed ;  though 
detailed  figures  must  be  omitted  for  want  of  room.  PI.  X.,  Fig. 
72,  shows  a  skeleton  sketch  of  such  a  design,  which  is  some- 
times adopted  on  account  of  the  stately  appearance  of  a  dome- 
like ceiling.     Here  let  ABCD  be  the  half  of  a  square  floor,  of 


STONE-CUTTING.  115 

which  the  circumscribed  circle,  of  radius  OB,  is  the  base  of  a 
hemisphere.  The  four  walls  of  the  room  will  then  be  bounded 
by  vertical  small  semicircles,  as  BC  —  A'Q'D'  ;  the  ceiling 
F'H'E',  within  the  circle  of  radius  OA  will  be  a  spherical  seg- 
ment ;  and  the  four  areas  like  ABI  will  be  covered  by  spherical 
gores,  shown  more  clearly  at  A"B"I",  in  the  elevation  made  on 
a  vertical  plane,  whose  ground  line  is  mq,  perpendicular  to  the 
diagonal  BO. 

128.  Two  joint-systems. — The  joints  of  the  spherical  surface 
may  then  be  either  (a)  horizontal  small  circles,  and  vertical 
meridians;  or  (F)  vertical  small  circles,  and  meridians,  all 
having  BO  — B"  for  a  common  diameter ;  the  beds  bounded  by 
the  small  circle  joints  being  conical  in  both  cases. 

Examples. —  1°.  Make  figure  69,  on  a  scale  of  ^L,  or  larger,  and  with  the  arch 
smaller  in  proportion. 

2°.  The  same  with  the  two  elevations  side  by  side. 

3°.  The  same,  with  the  axis  of  the  arch  coinciding  with  a  horizontal  diameter 
of  the  dome. 

4°.  Complete  the  projection  of  the  dome  on  V- 

5.  Construct  the  dome  with  pendentives,  Eig.  72,  in  detail  on  a  large  scale,  and 
by  each  joint-system. 

SPIRALS. 

129.  A  few  observations  on  the  spirals  found  in  the  next 
problem  are  here  added,  as  they  may  not  be  conveniently  acces- 
sible elsewhere. 

A  SPIRAL  is  a  plane  curve,  generated  by  a  point  which  has 
two  simultaneous  motions,  or,  more  precisely,  whose  actual 
motion  can  be  revolved  into  two  components  ;  one,  a  rotary  mo- 
tion, around  a  central  point  called  the  pole  ;  the  other,  a  radial 
motion,  outward  from  the  pole. 

130.  Illustrations.  The  spiral  of  Archimedes.  —  This  is  the 
simplest  of  all  the  spirals  ;  since  each  of  the  component  motions 


Fig.  11. 


116  STEREOTOMY. 

of  the  generatrix  is  uniform.  Thus  in  Fig.  11,  let  O  be  the 
pole,  and  OA,  the  initial  line,  so-called,  on  which  the  successive 
equal  increments  of  the  radial  movement  are  laid  off.  Then 
divide  any  circle,  having  O  for  its  centre,  into  equal  parts,  as  at 
1,  2,  3,  etc.,  and  make  0^  =  06;  0<?j  =  Oc;  0^  =  0 A,  etc., 

and  Oa^tf! tangent  to  OA  at  O,  will  be  a  spiral  of 

Archimedes.  The  distance  of  any  point  of  the  curve  from  the 
pole  is  called  its  radius  vector. 

In  this  example  the  circle  is  divided  in  16  equal  parts ;  hence, 
a  circle  of  a  radius,  which  we  will  call  0AX,  comprising  16  of  the 
parts  of  OA,  from  O,  would  be  divided  into  the  same  number 
of  parts  as  OA^  As  OA  and  any  fractional  part  of  the  circle 
of  radius  OA  may  be  divided  into  the  same  number  of  equal 
parts  there  may  be  an  infinite  variety  of  spirals  of  Archimedes. 

Let   the  — th  part   of   circle  OA  be  divided  into  the  same 
n       A 

number  of  parts  as  the  radius  OA.  Then,  calling  the  radius 
vector  =  r ;  OA  =  a,  and  the  arc,  as  A3,  corresponding  to  any 
radius  vector,  as  Oc,  =  0,  we  have,  directly  from  the  definition, 


r  : 
r. 

\a 

n 
a6 

.2ira; 

0 

.2-rra 

m 

.2tt 

whence  r  = = CI) 


which  is  the  general  equation  of  the  spiral  in  the  form  most 
convenient  for  use  in  drawing  tangents  to  it  by  the  method  of 
resultants. 

131.   Tangent  to  the  spiral  of  Archimedes.   Differentiating  (1) 

-r  =  -  2tt  ;  where  -r-,  or  is  the  ratio  of   the   rotary  and 

dr         n  dr  n  * 

the  radial  components  of  the  motion  of  the  generatrix,  the 

former  being  referred  to  the  circumference  of  the  circle  whose 

radius  is  a.     The  application  will  be  better  understood  by  an 

example. 

As  we  may  generally  make  m  =  1,  write  at  once  -r  =  — . 

Then  let  it  be  required  to  construct  the  tangent  at  5|,  in  Fig. 
12.  Here,  n  =  4,  since  i  of  the  circle  OA  is  divided  into  the 
same  number  of  parts  as  are  found  on  the  line  OA.  Then  lay 
off  on  Fbi  produced,  bYn  =  4,  on  any  convenient  scale ;  and 
2p  =  2tt,  by  the  same  scale,  on  the  tangent  at  2  to  the  circle  O  A ; 
reduce  the  rotary  component  as  estimated  with  the  radius  P2, 


STONE-CUTTING. 


117 


to  its  actual  value,  byS,  parallel  to  2p,  at  bv  by  drawing  the 
radius  Pp.  Then,  bxt,  the  diagonal  of  the  parallelogram  on  the 
components,  bin  and  b^s,  is  the  required  tangent  at  bx. 

n 1- 


Fig.  12. 
If  n  =  1,  PcZ,  and  the  circle  of  radius  Fd  will  be  divided  into 
the  same  number  of  equal  parts,    -5-  =  -j-'  and  2p  would  be 

4  times  2p,  or  ^w  would  have  been  called  1  instead  of  4. 

132.  The  subnormal  method.  —  Draw  PQ  perpendicular  to 
P51?  and  limited  at  Q  by  the  normal  ^Q.  Then  PQ  is  the  sub- 
normal.    Now  the  triangles  PQ5i  and  b^it  are  similar,  and  give, 

whence  PQ  =  b,n  X  -r-*- 

But  if  bxn  is  made  constant  for  each  point,  *2p  will  be  so  also ; 

and  hence,  as  we  see  from  the  figure,  bys  will  vary  as  PZ^ ;  that 

Vb 
is,  the  ratio  ~  will  be  constant.    Thus  PQ  is  constant.  Hence, 

having  any  one  tangent,  any  other  can  be  found  as  follows. 
Take,  for  example,  the  point  ex.  Draw  a  perpendicular,  PQx, 
not  shown,  to  its  radius  vector  Pg2;  at  P,  and  equal  to  PQ. 
Then  Q^  will  be  the  normal  at  eu  where  the  tangent  will  then 
be  perpendicular  to  Q^j. 

133.  The  tangentoid  spiral.  —  This  is  one  of  a  series  of 
curves  known  as  the  trigonometrical  spirals,  in  each  of  which 
the  radius  vector,  r,  is  some  trigonometrical  function  of  the 
angle,  (9,  between  it  and  the  fixed  initial  line. 


118 


STEREOTOMY. 


The  tangentoid  spiral  is  that  in  which  the  increments  of  the 
radius  vectors  are  equal,  or  proportional,  to  the  increments  of 
the  vectorial  angle,  6. 


a-* 

»/ 

.-     ::     i,     -   - 

a,  "■ 

-P-                ba/ 

- 

a-a 

p 

Fig.  13. 

Thus,  let  P,  Fig.  13,  be  the  pole,  and  Pa  =  a,  the  initial  line. 
Divide  the  circle  Pa  at  pleasure  as  at  1,  2,  etc.,  then  on  P&,  for 
example,  make  r  ==  PB  =  ah  =  Pa  X  the  tangent  of  BPa, 
and  B  will  be  a  point  of  a  tangentoid  spiral,  whose  equation, 
simply  expressing  the  construction,  is,  r  =  a.  tan  6  ;  or,  if  a  =  1, 
then  r  =  tan  6 ;  that  is,  r  is  equal  to  the  tangent  of  0.  But  if, 
having  drawn  either  a-J)x,  or  a2b2  parallel  to  ah,  we  should  make 
PBX  (not  shown),  =  a-}>x ;  or,  PB2  =  a2b2,  we  should  find  new 
forms  of  the  spiral ;  where  if  Pax  ==  ax  and  Pa2  =  a2,  then- 
equations  would  be  r  =  av  tan  6 ;  and  r  =  a2.  tan  0,  and  r 
would  be  proportional  to  tan  0. 

134.  Initial  line  a  secant.  —  In  Fig.  13,  the  initial  line,  Pa, 
is  evidently  tangent  to  the  spiral  at  the  pole,  P.  This  is  not 
always  so. 

K 


\. 

a 

c                h 

/   \ 

\    a. 

E^ 

~pr\^ 

«<j/' 

/'' ': 

,-;/^H 

*K 

/.--— 

ci  /'       ! 

/ 

/ 

rkn, 

i     ,-"' 

d  \i     4 

In,/ 

\ 

M  A'  / 

/  /k 

/Ii 

iK\ 

A\\ 

/     /*'/ 

Fig.  14. 


STONE-CUTTING.  119 

Thus,  Fig.  14,  let  Ea  and  EP  include  any  angle  whatever, 
and  let  them  be  divided  proportionally  by  parallels  as  Aa,  Bb 
etc.,  including  PL.  Then  arcs  from  A,  B,  etc.,  will  meet  the 
corresponding  radials,  Pa,  P5,  etc.,  in  points  ax,bt,  etc.,  of  a 
curve  which  will  still  be  a  tangentoid  spiral.  For  the  incre- 
ments, CD,  CB,  etc.,  of  the  radius  vector,  r,  estimated  from  the 
circle  Ccl5  where  Pcxc  is  perpendicular  to  Ea,  are  proportional 
to  the  increments,  cd,  cb,  etc.,  of  the  tangents  of  6,  where  8  is 
estimated  from  Pc. 

Evidently  P  is  the  pole,  and  PL  the  tangent  at  P. 

Then,  in  order  to  write  r  =  tan  6 ;  make  ML  =  Pc: ;  and 
take  6  =  MPL.     Then  ML  =  tan  6  ;  and  MP  =  r  =  1. 

Also  the  tangent  increment  Mm  =  the  radial  increment 
cji  =  CA,  where  Pro  is  the  radius  vector  drawn  through  the 

point  av     Likewise,  Mn  =  CE,  etc. 

dv 
135.   The  tangent  line.  —  From  r  =  tan  6,  -75-=  sec2  6  = 

sec20 
l2    ' 

Now  at  «!,  for  example,  Pra  =  sec  8,  and  PM  =  1.  Then 
a  third  proportional  (x)  to  PM,  and  Pm,  will  be  the  longer 
side  of  a  rectangle  ;  equal  to  (Pm)2,  and  whose  other  side  equals 
PM.  ThatisPMxz=(Pra)2-  Butthe  figures  VMXx  and  PM2 
(=  l2)  having  the  common  altitude  PM,  are  to  each  other  as 

x  and  PM.     That  is,  Jq  =  -^~  =  (m2)=  ^^  =  _ . 

Hence,  make  ^H  =  the  3d  proportional,  x,  for  the  radial 
component  of  the  motion  of  the  generatrix ;  and  M&  =  MP  =  1 
for  the  rotatory  component,  referred  to  the  line  on  which  tan  6  is 
estimated,  and  which  will  be  reduced  by  the  line  Vk  to  its 
value,  Ml5  for  the  circle  of  radius  Yav  Then,  making  HK  = 
hku  K«j  is  the  required  tangent  at  ax. 

We  now  close  with  the  following,  which,  besides  its  interest 
as  a  structure,  embraces,  as  appropriate  for  a  final  problem, 
representatives  of  all  the  four  classes  of  surfaces  which  form 
the  main  divisions  of  descriptive  geometry,  and  of  its  applica- 
tion in  this  volume. 


120  STEEEOTOMY. 


Problem  XX. 


The  annular  and  radiant  groined  arch. 

136.  Supposed  conditions.  —  Suppose  that  a  building  for  a 
private  library  or  cabinet,  or,  if  of  suitable  dimensions,  for  a 
locomotive  engine  bouse,  is  to  consist  of  a  gallery,  annular  in 
plan,  and  inclosing  a  central  circular  area.  Also  let  the 
gallery  be  divided  into  seven  compartments  by  arches,  whose 
straight  elements  shall  radiate  from  a  vertical  line  at  the  centre 
of  the  central  area. 

I.  The  Projections.  1°.  — Let  a  vertical  line  at  O,  Plate  X., 
Fig.  73,  be  the  axis  of  four  concentric  vertical  cylinders  gener- 
ated by  the  revolution  about  this  axis  of  vertical  lines  at  A, 
a,  5,  and  B,  where  OA  is  27  ft.  6  in.  ;  OB,  12  ft.  6  in.  ;  AB 
15  feet ;  and  ah  9  feet,  making  the  thickness  of  the  walls  Aa 
and  B5,  3  feet. 

Let  the  circular  gallery  between  the  walls  be  covered  by  an 
arch,  whose  intrados  is  the  half  annular  torus  generated  by  the 
revolution  of  the  vertical  semicircle  aob,  of  4  ft.  6  in.  radius, 
shown  at  ac2b  on  the  plane  V-  Let  this  gallery  be  divided  into 
seven  equal  compartments,  one  of  which  is  bounded  by  the  arc 
AD,  of  24.68  ft.,  and  the  radials,  OA  and  OD. 

This  arc,  AD,  may  be  accurately  determined  by  laying  off 

any  suitable  fractional  part,  as  —  of  it,  n  times.     Or,  from  a 

table  of  sides  of  inscribed  regular  polygons,  we  can  find  the 
length  (here  23.86  ft.)  of  the  chord  equal  to  one  side  of  the 
regular  polygon  (here  a  heptagon),  indicated  by  the  number 
of  compartments. 

Setting  off  each  way  from  A,  D,  etc.,  3  ft.  on  the  outer  cir- 
cumference, as  at  AE  and  AF,  and  drawing  the  radii,  as  OE 
and  OF,  we  have  the  piers,  as  shown  at  Ee/F,  and  HhH  ;  and 
the  area  GFIJ  which  is  to  be  covered  by  the  radiant  arch.  This 
arch  is  here  represented  as  closed  at  its  ends  by  twelve  inch 
walls  ;  which,  with  the  piers,  form  alcoves. 

2°.  The  intrados  of  the  radiant  arch  will,  in  any  case,  natur- 
ally be  a  right  conoid  (114),  extending  from  OF  to  OG,  and 
whose  springing  plane  will  be  the  same  as  that  of  the  annular 
arch;  viz.,  the  horizontal  plane  containing  the  diameter  ab. 
The  conoid  will  then  be  generated  by  the  straight  line  OF, 


Pi.vnr 


STONE-CUTTING.  121 

moving  so  as  to  be  parallel  to  the  springing  plane  as  a  plane 
director,  while  moving  upon  the  vertical  line  at  O,  and  some 
curve  o£  a  height  equal  to  oc2,  and  included  symmetrically  be- 
tween OF  and  OG. 

3°.  Two  systems.  —  At  this  point,  there  is  a  choice  between 
the  two  methods,  one  or  the  other  of  which  would  be  most 
naturally  chosen. 

First.  The  curved  directrix  may  be  the  ellipse  whose  trans- 
verse axis  is  the  chord  of  any  arc,  as  ooY,  or  GF,  included  be- 
tween OG  and  OF,  and  whose  semi- conjugate  axis  is  a  verti- 
cal, equal  to  oc2  at  the  middle  point  of  such  chord. 

Second.  The  curved  directrix,  may,  instead,  be  the  curve  of 
double  curvature,  formed  by  wrapping  upon  some  of  the  vertical 
cylinders,  as  DGA,  a  semi-ellipse  whose  transverse  axis  is 
the  true  length  of  the  corresponding  arc,  as  GF,  and  whose 
semi-conjugate  axis  equals  oc2.  as  before. 

4°.  Adopting  the  second  system.  —  KGiK^  is  the  curved 
directrix  of  the  conoidal  intrados ;  where  KG^  tangent  to 
GKF  at  K,  equals  the  arc  KG,  and  KKX  perpendicular  to  KGi 
equals  oc2.  The  method  of  concentric  circles  on  the  given 
axes,  is  adopted,  as  shown,  for  constructing  this  eclipse,  on  ac- 
count of  its  convenience  in  affording,  at  the  same  time,  an  easy 
construction  of  any  desired  tangents,  as  also  shown. 

5°.  The  groin  curves.  —  These  are  found  by  auxiliary  hori- 
zontal planes,  taken  for  convenience  though  the  inner  extremi- 
ties a1?  etc.,  of  the  radial  joints  of  the  annular  arch,  where  ac2b 
is  divided  here  into  five  equal  parts.  Each  plane  will  then 
cut  two  horizontal  circles,  centred  at  O,  from  the  torus,  and 
two  elements  from  the  conoid,  whose  intersections  with  the 
circles  will  be  four  points  of  the  groin  curves.  Thus  g,  f,  e,  /, 
are  the  points  determined  by  the  springing  plane  ;  cl  the  apex 
of  the  groin,  is  the  intersection  of  the  circle  Oo  with  the 
element  KO.  Then,  for  example,  making  b",  at  the  height, 
b'{x2,  =  bxq  ;  and  Kxx,  =  Kx2  =  KM,  the  two  circles  with 
radii,  Op  and  Oq,  will  intersect  the  corresponding  elements 
Oxx  and  OM,  at  the  four  points,  I,  Zx,  m,  and  mx  of  the  groin 
curves. 

6°.  Nature  of  the  horizontal  projections  of  the  groin  curves. — 
Each  of  those  just  found  is  an  arc  of  a  spiral  of  Archimedes 
(130),  for,  from  the  properties  of  ellipses  having  an  axis  in 
each  equal  (oc2  =  KKX)  we  have, 


122  STEREOTOMY. 

oa3  :  op  ::  KX2  :  Kx2 ; 

or,  by  the  substitution  of  equal  terms, 

n2n  :  m2m  ::  KN  :  KM. 

That  is,  the  increments,  n2n  and  m2m,  of  the  radius  vectors, 
are  proportional  to  the  corresponding  increments,  KN  and  KM 
of  the  arc  which  marks  their  angular  movement. 

The  pole  and  the  initial  line.  —  O,  the  given  intersection  of 
the  radius  vectors,  is  the  pole ;  and,  knowing  the  character  of 
the  curve,  find  a  fourth  proportional  to^  (=  «6)  GF  (=  2GXK) 
and  JO  (==  JO)  and  lay  it  off  from  G  to  the  left,  or  from  F  to 
the  right,  on  the  circle  OK,  and  points  will  be  found,  where 
the  radii  to  O  will  be  the  initial  lines  of  the  spirals,  fcxj  and 
gcxi,  respectively  ;  and  hence,  tangent  to  them  at  O. 

The  portions  of  these  spirals  beyond  ^  and/",  and  within  i 
and  j,  as  at  jCO,  being  projections  of  no  actual  lines  of  the 
structure,  are  called  parasites. 

7°.  The  tangentoid  system.  —  Turning  to  this  (3°)  for  a  mo- 
ment, for  comparison,  suppose  that  the  vertical  ellipse,  of  trans- 
verse axis,  gf,  and  vertical  semi-conjugate  axis  at  o2,  =  oe2,  had 
been  chosen  as  the  curved  directrix  of  the  conoid.  Then,  as  ox 
and  o2  are  at  equal  heights,  lines  parallel  to  oxo2,  as  yxy2,  would 
give  elements  ;  circular,  with  radius  Oyx,  through  yx  on  the 
torus,  and  straight  at  y20  on  the  conoid,  which  would  meet  as 
at  y,  a  point  of  the  curve  of  intersection  of  the  torus  and  the 
new  conoid;  a  curve  whose  horizontal  projection,  gycx,  etc., 
would  be  an  arc  of  a  tangentoid  spiral  (133).  For  evidently 
the  increments,  as  oxyx,  yxg,  etc.,  of  the  radius  vectors,  are  pro- 
portional to  the  increments,  as  o2y2,  y2g,  etc.,  of  the  tangents  of 
the  angles  made  by  these  radius  vectors  with  Oo2  as  an  initial 
line  (134). 

8°.  Section  and  joints  of  the  annular  arch. —  Let  the  figure 
AujVxWfrB  be  the  section  of  the  annular  arch.  By  revolution 
about  the  vertical  axis  at  O,  this  figure,  being  in  the  vertical 
plane  on  OA,  will  generate  the  volume  of  masonry,  covering 
the  annular  arch  ;  and  its  radial  lines,  as  axvx,  will  generate  the 
coursing  surfaces,  or  beds,  of  the  voussoirs.  These  beds  will 
obviously  be  conical  surfaces,  as  at  Wna^v,  all  having  the  ver- 
tical line  at  O  for  their  common  axis.  The  heads,  or  transverse 
joints,  as  at  YZ,  are  in  vertical  planes  through  O. 

9°.   The  joints   of  the   conoidal  arch.  —  The   bed  surfaces 


STONE-CUTTING.  123 

should,  strictly,  be  normal  to  the  conoidal  inti-ados,  along  the 
elements,  as  MMl5  which  are  the  coursing  joints.  They  will 
therefore  be  hyperbolic -paraboloids  (117). 

But  at  the  highest  element,  the  conoidal  surface  is  develop- 
able along  one  element,  KO,  since  a  horizontal  tangent  plane 
will  evidently  there  be  tangent  all  along  that  element.  Hence, 
the  normal  surface  on  KO  will  also  be  a  plane,  and  the  normal 
surfaces  on  elements  near  KO  will  therefore  be  very  nearly 
plane.     Hence,  the  bed  surface  on  NNX  may  properly  be  plane. 

10°.  Construction  of  the  warped  bed  on  MMX.  —  Assume  at 
V2  a  vertical  plane  of  projection,  L/T',  perpendicular  to  MM1 ; 
on  which  MMj  is  therefore  projected  in  the  point  m! ,  at  the 
height  \Jm'  =  pax  =  bxx2  on  Vi-  Drawing  the  tangent  b'{T',  as 
shown,  make  MT,  tangent  at  M,  equal  to  x2T",  and  by  project- 
ing T  at  T7,  m'T  will  be  its  vertical  projection.  Then  (116) 
TO  will  be  an  element  of  one  generation,  and  the  horizontal 
trace  of  the  tangent  hyperbolic  paraboloid,  generated  by  the 
motion,  parallel  to  H,  of  MO, upon  MT  —  m'T  and  O  —  L'M'. 
Hence,  mt,  mxu,  MiU,  and  Ss,  are  elements  of  the  other  gener- 
ation of  the  same  tangent  surface  ;  and  m't' ,  m'u',  etc.,  are  their 
vertical  projections. 

Taking,  now,  a  horizontal  plane,  M's'x,  at  a  height,  L'M', 
equal  to  vxqx ;  perpendiculars  m'W,  mV,  etc.,  to  m'T,  m't',  etc., 
and  limited  by  M.'s{,  will  be  those  elements  of  the  normal  hy- 
perbolic paraboloid,  which  are  the  lines  MT  —  m'T,  etc.,  re- 
volved 90°  about  MMx  —  m'  as  an  axis  (117).  Hence,  project 
R'  on  MT  at  R ;  r'  on  mt  at  r,  etc.,  and  MmM^rR  will  be 
the  horizontal  projection  of  the  required  normal  bed  on  MML. 

The  actual  limits  of  this  bed  are  at  its  intersections  mp2Q, 
and  w?iQi,  with  the  corresponding  conical  beds,  Qq^m,  and 
QiW?^,  of  the  annular  arch.  These  are  found  as  follows  :  As- 
same  any  intermediate  point  P  on  the  joint  axvx,  maker'V  at  the 
height  from  L'T',  equal  to  Fpx,  and  project  r"  at  r'"  on  MT; 
2'  at  z,  on  mt  ;  s"  at  s'",  etc.,  and  r'"zs'"  will  be  the  intersection 
of  the  normal  joint  on  MM1?  with  the  plane  r"s" ;  limited  at  p2 
by  p\p2,  the  corresponding  horizontal  section  of  the  torus. 
Therefore,  MRQp2m  is  a  definite  normal  joint.  M.xmlQ^kl  is 
another. 

11°.  The  outer  edge,  RRl5  of  the  tvarped  bed  on  MMy' is  a  hy- 
perbola.— This  we  know  from  the  properties  of  the  normal  sur- 
face, since  any  plane  which  cuts  the  intersection  of  the  two 


124  STEREOTOMY. 

plane  directors  cuts  the  surface  in  a  hyperbola ;  and  as  the  tan- 
gent hyperbolic  paraboloid  on  MMX,  revolves  90°  on  MMX  to 
become  the  normal  one,  its  former  plane  director,  of  the  gener- 
ation MMl5  which  is  H?  becomes  a  vertical  plane,  parallel  to 
MMj.  The  other  plane  director  is  perpendicular  to  MMX.  The 
intersection  of  the  two  is  thence  a  vertical  line,  which  accord- 
ingly cuts  the  horizontal  plane  M'si,  containing  the  curve  RRl5 
which  is  therefore  a  hyperbola. 

Otherwise  ;  by  direct  demonstration.  The  triangles  T'L'm' 
and  m'M'R',  and  other  like  pairs,  are  similar,  and  give 

L'T'  :  Wmr  ::  L'm'  :  M'R' 
Also,  L't'  :  Wm'  : :  L'm'  :  Mr' 

whence  L'T'  X  M'B/=  L't'  X  M¥. 

That  is,  MT  X  MR  =  mt  X  mr. 

Substituting  for  MT  and  mt,  the  proportionals  to  them,  MO 
and  mO,we  have 

MO  X  MR  =  raO  X  mr  ; 
which  may  be  written,  xy  =  x'y' ; 

if  the  curve  be  referred  to  OM,  and  a  perpendicular,  002,  to  it 
at  O,  as  axes. 

At  the  point  equidistant  from  the  axes, 

x'  =  y',  and  putting  x'y'  =  k2,  we  have  xy  =  Je2;  which  is  the 
equation  of  the  hyperbola,  referred  to  its  asymptotes. 

At  the  point  |l5  for  example,  x=  OS,  y=.  S?i. 

The  mean  proportional  to  these  is  Sl=&;  whence  2,  the 
vertex,  is  found,  as  shown,  on  the  transverse  axis,  02,  of  the 
curve,  which  bisects  the  angle,  SOOx,  between  the  asymptotes. 

This  being  here  a  right  angle,  the  hyperbola  is  of  the  form 
called  equilateral. 

12°.  Construction  of  an  approximate  plane-normal  joint,  on 
NNj.  —  The  point  N  is  symmetrical  with  Xj,  hence  the  tangent 
at  Xx  is  symmetrical  with  that  at  N.  Then  draw  the  tangent 
b'ik,  Make  Vi  tangent  to  the  cylinder  DKA  at  X15  then  Xja: 
(==  X2K)  will  be  the  horizontal  projection  of  this  tangent. 
But  b%k  being  straight,  and  oblique  to  the  elements,  as  Kk  of 
the  cylinder,  will,  when  wrapped  upon  the  cylinder  DKA,  be 
transformed  into  a  helix,  whose  projection,  and  that  of  its  tan- 
gent, on  Vi,  will  in  the  vicinity  of  k,  be  sensibly  b'^k  produced. 
Hence, -make  arc  Xjz"  =X1a;,  project  x"  on  b'{h  produced,  and 
x"~kx,  will  be  the  true  height  at  which  the  tangents  at  X^^) 
and  at  N,  will  pierce  the  plane  OK. 


STONE-CUTTING.  125 

This  found,  take  x'v'  for  the  ground  line  of  a  new  vertical 
plane  V3 ;  project  N  upon  it  at  the  height  a3n'  (=  a3a2),  and  x 
at  ki  at  the  height  x"kx ;  when  n'  Tcx  will  be  the  element  at  Nw/  of 
the  tangent  hyperbolic  paraboloid  on  NNX  ;  and  n'V  perpendic- 
ular to  it,  an  element  of  the  normal  hyperbolic  paraboloid  on 
NNl5  where  v'Y'  =vY.  Hence,  projecting  V  at  N2  (on  the 
tangent  at  N)  N2N",  parallel  to  NNl5  will  be  the  trace,  on  the 
horizontal  plane  V'V"  of  the  top  of  the  annular  arch,  of  the 
tangent  plane  at  ~Nn'  to  the  normal  hyperbolic  paraboloid ; 
that  is,  of  the  approximate  plane  joint. 

In  fact,  two  such  planes  should  have  been  similarly  made ; 
one  tangent  at  the  middle  point  of  Nw,  for  the  bed  NN2N'w  ; 
and  one  at  the  middle  point  of  N^j,  for  the  bed  N^N". 

II.  The  Directing  Instruments.  — After  the  previous  thorough 
working  up  of  all  the  lines  and  surfaces  composing  the  required 
structure,  these  can  now  be  summarily  described. 

Besides  Nos.  1  and  2  (7),  there  should  be,  for  the  stone 
"S^'Wna^qi,  for  example,  — 

No.  3.     The  pattern  l$"''N'vql,  of  the  horizontal  plane  top. 

No.  4.  The  pattern  of  the  cylindrical  back,  which,  to  the 
the  right  of  Qt,  is  temporary.  In  No.  4,  q3Q  =  q^N,N  ;  J46  == 
n{V',  both  in  length  and  direction;  544  =  X352';  4  5,  and  5b"s 
are  respectively  equal  to  the  like  spaces  on  the  plan  ;  and 
qsqi  =  V^i-     No.  5  is  a  pattern  of  the  end,  W1v1a1a2,  in  the  arch. 

No.  6  is  the  development  of  the  conical  bed,  Qq^pm.  The 
vertex  of  this  cone  is  found  after  revolution,  at  O',  the  inter- 
section of  00'  perpendicular  to  OA,  with  vxax  produced.  Hence 
with  O'  as  a  centre  draw  a1ms=pm,  and  v1Q1  =  ^1Q  ;  likewise 
VPl=pip2,  and  Qllvlalm3  will  be  the  required  pattern. 

No.  7  is  the  pattern  found  in  the  same  way  as  was  No.  6,  of 
the  conical  bed  Wva3n. 

No.  8,  which  could  exist  only  for  a  plane  bed,  is  the  pattern 
N5N44w,  of  the  plane  joint  N'"N'w4  ;  hence  N44  =  Y'n'. 

Nos.  9  and  10,  bevels  giving  respectively  the  angles  V1V«2, 
and  Ya2a^  will  be  useful  in  determining  the  relative  position  of 
the  top,  and  the  conical  bed  Wvazn ;  and  of  this  bed  and  the 
annular  intrados  ;  both  9  and  10,  being  held  in  meridian  planes 
of  the  torus. 

No.  11,  set  to  the  angle  n'Y  !n",  will,  in  like  manner,  serve 
to  fix  the  proper  position  of  N'N'"4w,  relative  to  the  plane  top. 


126  STEBEOTOMY. 

III.  Application.  —  Choosing  a  sufficient  block,  bring  it  pro- 
visionally to  the  form  of  a  prism,  whose  base  is  na3q24:,  except 
that  the  vertical  cylindrical  face  on  na  need  not  be  wrought. 

Then,  mark  the  intended  top  by  No.  3,  and  the  back  by  No. 

4.  The  lines  thus  given,  with  the  form  of  the  end  given  by  No. 

5,  will  guide  the  cutting  out  of  the  portion  Q,tqiq2. 

The  lower  conical  bed  can  then  be  directed  from  the  end, 
and  from  the  vertical  surface  on  VxVi,  and  shaped  by  No.  6. 
The  upper  conical  bed  will  then  be  determined  by  No.  9  and 
No.  7. 

The  remaining  work  is  obvious  enough,  the  construction  of 
No.  4  showing  the  ordinates  from  the  base  of  the  provisional 
prism  in  the  plane  453  to  the  edges  w4  and  naz. 

137.  St.  Gales'  Screw.  —  This  term  is  applied  to  circular 
stairs  with  solid  central  core,  or  column,  where  the  surface  seen 
overhead  in  ascending  the  stairs  is  a  double-curved  helicoidal 
surface,  such  as  would  be  generated  by  the  section,  AaV'BJ,  of 
an  annular  arch,  PI.  X.,  Fig.  73,  were  it  to  move  so  that  every 
point  of  it  should  describe  a  helix  about  the  vertical  axis  at  O  ; 
the  course  of  stones  generated  by  ~Qbbxvz  being  solid  with  the 
core.  The  coursing  surfaces  of  such  an  arch  would  be  oblique 
helicoids,  generated  by  axv^  etc.  ;  and  the  method  of  cutting 
of  the  stone  may  be  sufficiently  understood  from  the  cutting  of 
the  voussoirs  of  the  oblique  arch. 

Examples.  —  1°.  Make  the  full  construction  of  Fig.  73  on  the  tangentoid  system. 

2°.  Make  drawings  of  a  key-stone,  extending  both  ways  from  a  in  each  arch. 

3°.  Also  of  an  inner  pier  groin  stone,  as  Mi/niQi. 

4°.  Construct  the  plane  joints  indicated  in  12°. 

5°.  Make  the  necessary  illustrative  drawings  of  the  double-curved  arched  cover- 
ing of  circular  stairs,  known  as  the  St.  Giles'  screw  (137).  The  coursing  joints  are 
oblique  helicoids,  generated  by  radii  of  the  semicircle  ac^b,  Fig.  73.  The  heading 
joints  are  in  planes  perpendicular  to  the  helix  generated  by  the  highest  point,  &i, 
of  the  same  semicircle.  The  coursing  joints  thus  generated  are  not  quite  normal 
to  the  double-curved  arch  surface,  but  are  nearly  enough  so.  To  be  perfectly  so, 
they  should  be  generated  by  normals  to  that  intrados,  that  is,  by  lines,  not  only 
perpendicular  to  the  tangents  to  the  semicircle,  as  at  its  points  of  division,  etc., 
but  also  to  the  tangents  to  the  helices  at  the  same  points.  The  construction  would 
be  too  laborious,  and  thence  more  apt  to  be  inexact. 


PI.  IX 


M|-        v       '  t:        t 

if    ! 

V 

B'     Y' 

,   6'a'    i        i  v,  ■ 

10      Q  I         GA 


] 


^ 


lry\j 


